Bubbles Bad; Ripples Good

… Data aequatione quotcunque fluentes quantitates involvente fluxiones invenire et vice versa …

Category: classical analysis

Riemann-, Generalized-Riemann-, and Darboux-Stieltjes integrals

(The following is somewhat rough and may have typos.)

Let us begin by setting the notations and recalling what happens without the Stieltjes part.

Defn (Partition)
Let I be a closed interval. A partition P is a finite collection of closed subintervals \{I_\alpha\} such that

  1. P is finite;
  2. P covers I, i.e. \cup P = I;
  3. P is pairwise almost disjoint, i.e. for I_\alpha, I_\beta distinct elements of P, their intersection contains at most one point.

We write \mathscr{P} for the set of all partitions of I.

Defn (Refinement)
Fix I a closed interval, and P, Q two partitions. We say that P refines Q or that P \preceq Q if for every I_\alpha\in P there exists J_\beta \in Q such that I_\alpha \subseteq J_\beta.

Defn (Selection)
Given I a closed interval and P a partition, a selection \sigma: P \to I is a mapping that satisfies \sigma(I_\alpha) \in I_\alpha.

Defn (Size)
Given I a closed interval and P a partition, the size of P is defined as |P| = \sup_{I_\alpha \in P} |I_\alpha|, where |I_\alpha| is the length of the closed interval I_\alpha.

Remark In the above we have defined two different preorders on the set \mathscr{P} of all partitions. One is induced by the size: we say that P \leq Q if |P| \leq |Q|. The other is given by the refinement P\preceq Q. Note that neither are partial orders. (But that the preorder given by refinement can be made into a partial order if we disallow zero-length degenerate closed intervals.) Note also that if P\preceq Q we must have P \leq Q.

Now we can define the notions of integrability.

Defn (Integrability)
Let I be a closed, bounded interval and f:I \to \mathbb{R} be a bounded function. We say that f is integrable with integral s in the sense of

  • Riemann if for every \epsilon > 0 there exists P_0\in \mathcal{P} such that for every P \leq P_0 and every selection \sigma:P \to I we have
    \displaystyle \left| \sum_{I' \in P} f(\sigma(I')) |I'| - s \right| < \epsilon

  • Generalised-Riemann if for every \epsilon > 0 there exists P_0 \in \mathcal{P} such that for every P \preceq P_0 and every selection \sigma: P\to I we have
    \displaystyle \left| \sum_{I' \in P} f(\sigma(I')) |I'| - s \right| < \epsilon

  • Darboux if
    \displaystyle \inf_{P\in\mathscr{P}} \sum_{I' \in P} (\sup_{I'} f )|I'| = \sup_{P\in\mathscr{P}} \sum_{I' \in P} (\inf_{I'} f )|I'| = s

From the definition it is clear that “Riemann integrable” implies “Generalised-Riemann integrable”. Furthermore, we have clearly that for a fixed P
\displaystyle \sum_{I' \in P} (\inf_{I'} f) |I'| \leq \sum_{I' \in P} f(\sigma(I')) |I'| \leq \sum_{I' \in P} (\sup_{I'} f) |I'|
and that if P \preceq Q we have
\displaystyle \sum_{I' \in Q} (\inf_{I'} f) |I'| \leq \sum_{I' \in P} (\inf_{I'} f) |I'| \leq \sum_{I' \in P} (\sup_{I'} f) |I'| \leq \sum_{I' \in Q} (\inf_{I'} f) |I'|
so “Darboux integrable” also implies “Generalised-Riemann integrable”. A little bit more work shows that “Generalised-Riemann integrable” also implies “Darboux integrable” (if the suprema and infima are obtained on the intervals I', this would follow immediately; using the boundedness of the intervals we can find \sigma such that the Riemann sum approximates the upper or lower Darboux sums arbitrarily well.

The interesting part is the following
Darboux integrable functions are Riemann integrable. Thus all three notions are equivalent.

Proof. Let P, Q be partitions. Let |P| \leq \inf_{I'\in Q, |I'| \neq 0} |I'|, and let m be the number of non-degenerate subintervals in Q. We have the following estimate
\displaystyle   \sum_{I'\in Q} (\inf_{I'} f) |I'| - (m-1) |P| (\sup_I 2|f|) \leq \sum_{J'\in P} f(\sigma(J')) |J'| \leq \sum_{I'\in Q} (\sup_{I'} f) |I'| + (m-1) |P| (\sup_I 2|f|)
The estimate follows by noting that “most” of the J'\in P will be proper subsets of I'\in Q, and there can be at most m-1 of the J' that straddles between two different non-degenerate sub-intervals of Q. To prove the theorem it suffices to choose first a Q such that the upper and lower Darboux sums well-approximates the integral. Then we can conclude for all P with |P| sufficiently small the Riemann sum is almost controlled by the Q-Darboux sums. Q.E.D.

Now that we have recalled the case of the usual integrability. Let us consider the case of the Stieltjes integrals: instead of integrating against \mathrm{d}x, we integrate against \mathrm{d}\rho, where \rho is roughly speaking a “cumulative distribution function”: we assume that \rho:I \to \mathbb{R} is a bounded monotonically increasing function.

The definition of the integrals are largely the same, except that at every step we replace the width of the interval |I'| by the diameter of \rho(I'), i.e. \sup_{I'} \rho - \inf_{I'} \rho. The arguments above immediately also imply that

  • “Riemann-Stieltjes integrable” implies “Generalised-Riemann-Stieltjes integrable”
  • “Darboux-Stieltjes integrable” implies “Generalised-Riemann-Stieltjes integrable”
  • “Generalised-Riemann-Stieltjes integrable” implies “Darboux-Stientjes integrable”

However, Darboux-Stieltjes integrable functions need not be Riemann-Stieltjes integrable. The possibility of failure can be seen in the proof of the theorem above, where we used the fact that |P| is allow to be made arbitrarily small. The same estimate, in the case of the Stieltjes version of the integrals, has |P| replaced by \sup_{J'\in P} (\sup_{J'} \rho - \inf_{J'} \rho), which for arbitrary partitions need to shrink to zero. To have a concrete illustration, we give the following:

Let I = [0,1]. Let \rho(x) = 0 if x < \frac12 and 1 otherwise. Let f(x) = 0 if x \leq \frac12 and 1 otherwise. Let Q_0 be the partition \{ [0,\frac12], [\frac12,1]\}. We have that
\displaystyle \sum_{I'\in Q_0} (\sup_{I'} f) (\sup_{I'} \rho - \inf_{I'} \rho) = 0 \cdot (1 - 0) + 1\cdot (1 - 1) = 0
\displaystyle \sum_{I'\in Q_0} (\inf_{I'} f) (\sup_{I'} \rho - \inf_{I'} \rho) = 0 \cdot (1-0) + 0 \cdot(1-1) = 0
so we have that in particular the pair (f,\rho) is Darboux-Stieltjes integrable with integral 0. However, let k be any odd integer, consider the partition P_k of [0,1] into k equal portions. Depending on the choice of the selection \sigma, we see that the sum can take the values
\displaystyle \sum_{I'\in P_k} f(\sigma(I')) (\sup_{I'} \rho - \inf_{I'}\rho) = f(\sigma([\frac12 - \frac1{2k},\frac12 + \frac1{2k}])) (1 - 0) \in \{0,1\}
which shows that the Riemann-Stieltjes condition can never be satisfied.

The example above where both f and \rho are discontinuous at the same point is essentially sharp. A easy modification of the previous theorem shows that
If at least one of f,\rho is continuous, then Darboux-Stieltjes integrability is equivalent to Riemann-Stieltjes integrability.

Remark The nonexistence of Riemann-Stieltjes integral when f and g has shared discontinuity points is similar in spirit to the idea in distribution theory where whether the product of two distributions is well-defined (as a distribution) depends on their wave-front sets.

… and scattering a quantum particle

In the previous post we shot a classical particle at a potential barrier. In this post we shoot a quantum particle.

Whereas the behaviour of the classical particle is governed by Newton’s laws (where the external force providing the acceleration is given as minus the gradient of the potential), we allow our quantum particle to be governed by the Klein-Gordon equations.

  • Mathematically, the Klein-Gordon equation is a partial differential equation, whereas Newton’s laws form ordinary differential equations. A typical physical interpretation is that the state space of quantum particles are infinite dimensional, whereas the phase space of physics has finite dimensions.
  • Note that physically the Klein-Gordon equation was designed to model a relativistic particle, while in the previous post we used the non-relativistic Newton’s laws. In some ways it would’ve been better to model the quantum particle using Schroedinger’s equation. I plead here however that (a) qualitatively there is not a big difference in terms of the simulated outcomes and (b) it is more convenient for me to use the Klein-Gordon model as I already have a finite-difference solver for hyperbolic PDEs coded in Python on my computer.

To model a particle, we set the initial data to be a moving wave packet, such that at the initial time the solution is strongly localized and satisfies \partial_t u + \partial_x u = 0. Absent the mass and potential energy terms in the Klein-Gordon equation (so under the evolution of the free wave equation), this wave packet will stay coherent and just translate to the right as time goes along. The addition of the mass term causes some small dispersion, but the mass is chosen small so that this is not a large effect. The main change to the evolution is the potential barrier, which you can see illustrated in the simulation.

The video shows 8 runs of the simulation with different initial data. Whereas in the classical picture the initial kinetic energy is captured by the initial speed at which the particle is moving, in the quantum/wave picture the kinetic energy is related to the central frequency of your wave packet. So each of the 8 runs have increasing frequency offset that represents increasing kinetic energy. The simulation has two plots, the top shows the square of the solution itself, which gives a good indication of where physically the wave packet is located. The bottom shows a normalized kinetic energy density (I have to include a normalization since the kinetic energy of the first and last particles differ roughly 10 fold).

One notices that in the first two runs, the kinetic energy is sufficiently small that the particle mostly bounces back to the left after hitting the potential.

For the third and fourth runs (frequency shift \sqrt{2} and \sqrt{3} respectively) we see that while a significant portion of the particle bounces back, a noticeable portion “tunnels through” the barrier: this caused by a combination of the quantum tunneling phenomenon and the wave packet form of the initial data.

The phenomenon of quantum tunneling manifests in that all non-zero energy waves will penetrate a finite potential barrier a little bit. But the amount of penetration decays to zero as the energy of the wave goes to zero: this is known as the semiclassical regime. In the semiclassical limit it is known that quantum mechanics converge toward classical mechanics, and so in the low-energy limit we expect our particle to behave like a classical particle and bounce off. So we see that naturally increasing the energy (frequency) of our wave packet we expect more of the tunneling to happen.

Further, observe that by shaping our data into a wave packet it necessarily contains some high frequency components (due to Heisenberg uncertainty principle); high frequency, and hence high energy components do not “see” the potential barrier. Even in the classical picture high energy particles would fly over the potential barrier. So for wave packets there will always be some (perhaps not noticeable due to the resolution of our computation) leakage of energy through the potential barrier. The quantum effect on these high energy waves is that they back-scatter. Whereas the classical high energy particles just fly directly over the barrier, a high energy quantum particle will leave some parts of itself behind the barrier always. We see this in the sixth and seventh runs of the simulation, where the particle mostly passes through the barrier, but a noticeable amount bounces off in the opposite direction.

In between during the fifth run, where the frequency shift is 2, we see that the barrier basically split the particle in two and send one half flying to the right and the other half flying to the left. Classically this is the turning point between particles that go over the bump and particles that bounces back, and would be the case (hard to show numerically!) where a classical particle comes in from afar with just enough energy that it comes to a half at the top of the potential barrier!

And further increasing the energy after the seventh run, we see in the final run a situation where only a negligible amount of the particle scatters backward with almost all of it passing through the barrier unchanged. One interesting thing to note however is that just like the case of the classical particle, the wave packet appears to “slow down” a tiny bit as it goes over the potential barrier.

An optimization problem: variation

Examining the theorem proven in the previous post, we are led naturally to ask whether there are higher order generalizations.

Question: Let f \in C^{k}([-1,1]) with f^{(k)} > 0. What can we say about the minimizer of C = \int_{-1}^1 |f(x) - p(x)|~\mathrm{d}x where p ranges over degree k-1 polynomials?

It is pretty easy to see that we expect p to intersect f at the maximum number of points, which is k. We label those points x_1, \ldots, x_k and call x_0 = -1 and x_{k+1}= 1. Then the cost function can be written as
\displaystyle C = \sum_{j = 0}^k (-1)^j \int_{x_j}^{x_{j+1}} f(x) - p(x; x_1, \ldots, x_k) ~\mathrm{d}x
Since we know that values of p at the points x_1, \ldots, x_k we can write down the interpolation polynomial explicitly using Sylvester’s formula:
\displaystyle p = \sum_{j = 1}^k \left( \prod_{1 \leq m \leq k, m\neq j} \frac{x - x_m}{x_j - x_m} \right) f(x_j) = \sum L_j(x; x_1, \ldots, x_k) f(x_j)

The partial derivatives are now
\displaystyle \partial_n C = \sum_{j = 0}^k (-1)^{j+1} \int_{x_j}^{x_{j+1}} \partial_n p(x; x_1, \ldots, x_k) ~\mathrm{d}x
It remains to compute \partial_n p for 1 \leq n \leq k. We observe that when n \neq j
\displaystyle \partial_n L_j = - \frac{1}{x - x_n} L_j + \frac{1}{x_j - x_n} L_j
and also
\displaystyle \partial_n L_n = - \left( \sum_{1\leq m \leq k, m\neq n} \frac{1}{x_n - x_m} \right) L_n
\displaystyle \partial_n p =  \sum_{j \neq n} \frac{x-x_j}{(x_j - x_n)(x - x_n)} L_j f(x_j) + L_n f'(x_n) - \left( \sum_{1\leq m \leq k, m\neq n} \frac{1}{x_n - x_m} \right) L_n f(x_n)
Now, we observe that
\displaystyle \frac{x - x_j}{x - x_n} L_j =   - \left( \prod_{m \neq n,j} \frac{x_n - x_m}{x_j - x_m} \right)  L_n
so after some computation we arrive at
\displaystyle \partial_n p = L_n(x) \cdot \left[ f'(x_n) - \sum_{j \neq n} \frac{1}{x_j - x_n} \left(\left( \prod_{m \neq j,n}\frac{x_n - x_m}{x_j - x_m}\right)f(x_j) - f(x_n) \right)\right]
which we can further simplify to
\displaystyle \partial_n p = L_n(x) \cdot \left( f'(x_n) - p'(x_n)\right)
Now, since f and p cross transversely at x_n, the difference of their derivatives is non-zero. (This harks back to our assumption that f^{(k)} > 0.) So we are down, as in the case where k = 2, to equations entirely independent of f.

More precisely, we see that the stationarity condition becomes the choice of x_1, \ldots, x_k such that the integrals
\displaystyle \sum_{j = 0}^k (-1)^{j} \int_{x_j}^{x_{j+1}} L_n(x)  ~\mathrm{d}x = 0
for each n. Since L_n form a basis for the polynomials of degree at most k-1, we have that the function
\chi(x) = (-1)^j \qquad x \in (x_j, x_{j+1})
is L^2 orthogonal to every polynomial of degree at most k-1. So in particular the x_j are solutions to the following system of equations
x_0 = -1, \qquad x_{k+1} = 1
\sum_{j = 0}^k (-1)^j \left[ x_{j+1}^d - x_{j}^d \right] = 0 \qquad \forall d \in \{1, \ldots, k\}

From symmetry considerations we have that x_j = - x_{k+1 - j}. This also kills about half of the equations. For the low k we have

  1. \{ 0\}
  2. \{ -1/2, 1/2\}
  3. \{-1/2, 0, 1/2\}
  4. \{ (\pm 1 \pm \sqrt{5})/4 \}
  5. \{ 0, \pm\frac12, \pm \frac{\sqrt{3}}2 \}

An optimization problem: theme

Let’s start simple:

Question 1: What is the linear function \ell(x) that minimizes the integral \int_{-1}^1 |x^2  + x - \ell(x)| ~\mathrm{d}x? In other words, what is the best linear approximation of x^2 + x in the L^1([-1,1]) sense?

This is something that can in principle be solved by a high schooler with some calculus training. Here’s how one solution may go:

Solution: All linear functions take the form \ell(x) = ax + b. The integrand is equal to x^2 + x - \ell(x) when x^2 + x \geq \ell(x), and \ell(x) - x - x^2 otherwise. So we need to find the points of intersection. This requires solve x^2 + x - ax - b = 0, which we can solve by the quadratic formula. In the case where x^2 + x - a x - b is signed, we see that changing b we can make the integrand strictly smaller, and hence we cannot attain a minimizer. So we know that at the minimizer there must exist at least one root.

Consider the case where there is only one root in the interval (counted with multiplicity), call the root x_0. We have that the integral to minimize is equal to
\displaystyle \left| \int_{-1}^{x_0} x^2 + x - ax - b~\mathrm{d}x \right| + \left| \int_{x_0}^1 x^2 + x - a x - b ~\mathrm{d}x \right|
each part of which can be computed explicitly to obtain
\displaystyle \left| \frac13 x_0^3  + \frac13 + \frac{1-a}{2} x_0^2 - \frac{1-a}{2} - b x_0 - b \right| + \left| \frac13 - \frac13 x_0^3 + \frac{1-a}{2} - \frac{1-a}{2} x_0^2 - b + b x_0\right|
Since we know that the two terms comes from integrands with different signs, we can combine to get
\displaystyle \left| \frac23 x_0^3 + (1-a) x_0^2 - (1-a) - 2b x_0 \right|
as the integrand. Now, we cannot just take the partial derivatives of the above expression with respect to a,b and set that to zero and see what we get: the root x_0 depends also on the parameters. So what we would do then is to plug in x_0 using the expression derived from the quadratic formula, x_0 = \frac{1}{2} \left( a - 1 \pm \sqrt{ (1-a)^2 + 4b}\right), and then take the partial derivatives. Before that, though, we can simplify a little bit: since x_0^3 + (1-a) x_0^2 - b x_0 = 0 from the constraint, the quantity to minimize is now
\displaystyle \left| - \frac13 x_0^3 - (1-a) - b x_0 \right|
A long computation taking the \partial_b now shows that necessarily x_0 = 0, which implies that b = 0. But for the range of a where there is only one root in the interval, the quantity does not achieve a minimum. (The formal minimizer happens at a = 1 but we see for this case the integrand of the original cost function is signed.

So we are down to the case where there are two roots in the interval. Now we call the roots x_+ and x_-, and split the integral into
\displaystyle \left| \int_{-1}^{x_-} x^2 + x - ax - b~\mathrm{d}x  - \int_{x_-}^{x_+} x^2 + x - ax - b~\mathrm{d}x + \int_{x_+}^1 x^2 + x - ax - b~\mathrm{d}x \right|
and proceed as before. (The remainder of the proof is omitted; the reader is encouraged to carry out this computation out by hand to see how tedious it is.) Read the rest of this entry »

Bessaga’s converse to the contraction mapping theorem

In preparing some lecture notes for the implicit function theorem, I took a look at Schechter’s delightfully comprehensive Handbook of Analysis and its Foundations (which you can also find on his website), and I learned something new about the Banach fixed point theorem. To quote Schechter:

… although Banach’s theorem is quite easy to prove, a longer proof cannot yield stronger results.

I will write a little bit here about a “converse” to the Banach theorem due to Bessaga, which uses a little bit of help from the Axiom of Choice.

Read the rest of this entry »

Products and expectation values

Let us start with an instructive example (modified from one I learned from Steven Landsburg). Let us play a game:

I show you three identical looking boxes. In the first box there are 3 red marbles and 1 blue one. In the second box there are 2 red marbles and 1 blue one. In the last box there is 1 red marble and 4 blue ones. You choose one at random. What is …

  • The expected number of red marbles you will find?
  • The expected number of blue marbles you will find?
  • The expected number of marbles, irregardless of colour, you will find?
  • The expected percentage of red marbles you will find?
  • The expected percentage of blue marbles you will find?

Answer below the cut… Read the rest of this entry »

Whitney extension and counterexample to Sard’s theorem

In a first course in differential geometry/topology we are often taught the following version of Sard’s theorem:

Theorem (Sard, smooth case)
Let M and N be (finite dimensional) smooth manifolds, and let f:M\to N be a smooth (infinitely differentiable) map. Let S\subset N be the set of critical values, that is, for every y \in S there exists x \in f^{-1}(y) such that f'(x) is not surjective. Then S has measure zero.

It turns out that Sard in his 1942 paper proved something stronger.

Theorem (Sard)
Let f:\mathbb{R}^m \supset M\to N\subset \mathbb{R}^n be a mapping of class C^k (all partial derivatives up to order k exists and are continuous) for k \geq 1. Let S\subset N be defined as above. And if either

  1. m \leq n; or
  2. m > n and k \geq m - n + 1,

then S has measure 0.

Over at MathOverflow, Sergei Ivanov has shown that in the case m = n = 1, the statement can be modified so that C^1 is not necessary.

In this post, we will describe counterexamples in the case m > n if the condition on k is not satisfied. We will also sketch a natural generalisation of Ivanov’s argument for higher dimensions. Read the rest of this entry »

Mariş’s Theorem

During a literature search (to answer my question concerning symmetries of “ground states” in variational problem, I came across a very nice theorem due to Mihai Mariş. The theorem itself is, more than anything else, a statement about the geometry of Euclidean spaces. I will give a rather elementary write-up of (a special case of) the theorem here. (The proof presented here can equally well be applied to get the full strength of the theorem as presented in Maris’s paper; I give just the special case for clarity of the discussion.)

Read the rest of this entry »

Continuity of the infimum

Just realised (two seeks ago, but only gotten around to finish this blog posting now) that an argument used to prove a proposition in a project I am working on is wrong. After reducing the problem to its core I found that it is something quite elementary. So today’s post would be of a different flavour from the ones of recent past.

Question Let X,Y be topological spaces. Let f:X\times Y\to\mathbb{R} be a bounded, continuous function. Is the function g(x) = \inf_{y\in Y}f(x,y) continuous?

Intuitively, one may be tempted to say “yes”. Indeed, there are plenty of examples where the answer is in the positive. The simplest one is when we can replace the infimum with the minimum:

Example Let the space Y be a finite set with the discrete topology. Then g(x) = \min_{y\in Y} f(x,y) is continuous.
Proof left as exercise.

But in fact, the answer to the question is “No”. Here’s a counterexample:

Example Let X = Y = \mathbb{R} with the standard topology. Define

\displaystyle f(x,y) = \begin{cases} 1 & x > 0 \\ 0 & x < -e^{y} \\ 1 + x e^{-y} & x\in [-e^{y},0]  \end{cases}

which is clearly continuous. But the infimum function g(x) is roughly the Heaviside function: g(x) = 1 if x \geq 0, and g(x) = 0 if x < 0.

So what is it about the first example that makes the argument work? What is the different between the minimum and the infimum? A naive guess maybe that in the finite case, we are taking a minimum, and therefore the infimum is attained. This guess is not unreasonable: there are a lot of arguments in analysis where when the infimum can be assumed to be attained, the problem becomes a lot easier (when we are then allowed to deal with a minimizer instead of a minimizing sequence). But sadly that is not (entirely) the case here: for every x_0, we can certainly find a y_0 such that f(x_0,y_0) = g(x_0). So attaining the infimum point-wise is not enough.

What we need, here, is compactness. In fact, we have the following

Theorem If X,Y are topological spaces and Y is compact. Then for any continuous f:X\times Y\to\mathbb{R}, the function g(x) := \inf_{y\in Y} f(x,y) is well-defined and continuous.

Proof usually proceeds in three parts. That g(x) > -\infty follows from the fact that for any fixed x\in X, f(x,\cdot):Y\to\mathbb{R} is a continuous function defined on a compact space, and hence is bounded (in fact the infimum is attained). Then using that the sets (-\infty,a) and (b,\infty) form a subbase for the topology of \mathbb{R}, it suffices to check that g^{-1}((-\infty,a)) and g^{-1}((b,\infty)) are open.

Let \pi_X be the canonical projection \pi_X:X\times Y\to X, which we recall is continuous and open. It is easy to see that g^{-1}((-\infty,a)) = \pi_X \circ f^{-1}((-\infty,a)). So continuity of f implies that this set is open. (Note that this part does not depend on compactness of Y. In fact, a minor modification of this proof shows that for any family of upper semicontinuous functions \{f_c\}_C, the pointwise infimum \inf_{c\in C} f_c is also upper semicontinuous, a fact that is very useful in convex analysis. And indeed, the counterexample function given above is upper semicontinuous.)

It is in this last part, showing that g^{-1}((b,\infty)) is open, that compactness is crucially used. Observe that g(x) > b \implies f(x,y) > b~ \forall y. In other words g(x) > b \implies \forall y, (x,y) \in f^{-1}((b,\infty)) an open set. This in particular implies that \forall x\in g^{-1}((b,\infty)) \forall y\in Y there exists a “box” neighborhood U_{(x,y)}\times V_{(x,y)} contained in f^{-1}((b,\infty)). Now using compactness of Y, a finite subset \{(x,y_i)\} of all these boxes cover \{x\}\times Y. And in particular we have

\displaystyle \{x\}\times Y \subset \left(\cap_{i = 1}^k U_{(x,y_i)}\right)\times Y \subset f^{-1}((b,\infty))

and hence g^{-1}((b,\infty)) = \cup_{x\in g^{-1}((b,\infty))} \cap_{i = 1}^{k(x)} U_{x,y_i} is open. Q.E.D.

One question we may ask is how sharp is the requirement that Y is compact. As with most things in topology, counterexamples abound.

Example Let Y be any uncountably infinite set equipped with the co-countable topology. That is, the collection of open subsets are precisely the empty set and all subsets whose complement is countable. The two interesting properties of this topology are (a) Y is not compact and (b) Y is hyperconnected. (a) is easy to see: let C be some countably infinite subset of Y. For each c\in C let U_c = \{c\}\cup (Y\setminus C). This forms an open cover with not finite sub-cover. Hyperconnected spaces are, roughly speaking, spaces in which all open nonempty sets are “large”, in the sense that they mutually overlap a lot. In particular, a continuous map from a hyperconnected space to a Hausdorff space must be constant. In our case we can see this directly: suppose h:Y\to \mathbb{R} is a continuous map. Fix y_1,y_2\in Y. Let N_{1,2}\subset \mathbb{R} be open neighborhoods of f(y_{1,2}). Since h is continuous, h^{-1}(N_1)\cap h^{-1}(N_2) is open and non-empty (by the co-countable assumption). Therefore N_1\cap N_2\neq \emptyset for any pairs of neighborhoods. Since \mathbb{R} is Hausdorff, this forces h to be the constant map. This implies that for any topological space X, a continuous function f:X\times Y\to\mathbb{R} is constant along Y, and hence for any y_0\in Y, we have \inf_{y\in Y} f(x,y) =: g(x) = f(x,y_0) is continuous.

One can try to introduce various regularity/separation assumptions on the spaces X,Y to see at what level compactness becomes a crucial requirement. As an analyst, however, I really only care about topological manifolds. In which case the second counterexample up top can be readily used. We can slightly weaken the assumptions and still prove the following partial converse in essentially the same way.

Theorem Let X be Tychonoff, connected, and first countable, such that X contains a non-trivial open subset whose closure is not the entire space; and let Y be paracompact, Lindelof. Then if Y is noncompact, there exists a continuous function f:X\times Y\to\mathbb{R} such that \inf_{y\in Y}f:X\to \mathbb{R} is not continuous.

Remark Connected (nontrivial) topological manifolds automatically satisfy the conditions on X and Y except for non-compactness. The conditions given are not necessary for the theorem to hold; but they more or less capture the topological properties used in the construction of the second counterexample above.

Remark If X is such that every open set’s closure is the entire space, we must have that it is hyperconnected (let C\subset X be a closed set. Suppose D\subset X is another closed set such that C\cup D = X. Then C\subset D^c and vice versa, but D^c is open, so C = X. Hence X cannot be written as the union of two proper closed subsets). And if it is Tychonoff, then X is either the empty-set or the one-point set.

Lemma For a paracompact Lindelof space that is noncompact, there exists a countably infinite open cover \{U_k\} and a sequence of points y_k \in U_k such that \{y_k\}\cap U_j = \emptyset if j\neq k.

Proof: By noncompactness, there exists an open cover that is infinite. By Lindelof, this open cover can be assumed to be countable, which we enumerate by \{V_k\} and assume WLOG that \forall k, V_k \setminus \cup_{j =1}^{k-1} V_j \neq \emptyset. Define \{U_k\} and \{y_k\} inductively by: U_k = V_k \setminus \cup_{j = 1}^{k-1} \{ y_j\} and choose y_k \in U_k \setminus \cup_{j=1}^{k-1}U_j.

Proof of theorem: We first construct a sequence of continuous functions on X. Let G\subset X be a non-empty open set such that its closure-complement H = (\bar{G})^c is a non-empty open set (G exists by assumption). By connectedness \bar{G}\cap \bar{H} \neq \emptyset, so we can pick x_0 in the intersection. Let \{x_j\}\subset H be a sequence of points converging to x_0, which exists by first countability. Using Tychonoff, we can get a sequence of continuous functions f_jon X such that f_j|_{\bar{G}} = 0 and f_j(x_j) = -1.

On Y, choose an open cover \{U_k\} and points \{y_k\} per the previous Lemma. By paracompactness we have a partition of unity \{\psi_k\} subordinate to U_k, and by the conclusion of the Lemma we have that \psi_k(y_k) = 1. Now we define the function

\displaystyle f(x,y) = \sum_{k} f_k(x)\psi_k(y)

which is continuous, and such that f|_{\bar{G}\times Y} = 0. But by construction \inf_{y\in Y}f(x,y) \leq f(x_k,y_k) = f_k(x_k) = -1, which combined with the fact that x_k \to x_0 \in \bar{G} shows the desired result. q.e.d.

Conway’s Base 13 Function

(N.b. Credit where credit’s due: I learned about this function from an answer of Robin Chapman’s on MathOverflow, and its measurability from Noah Stein.)

Conway’s base 13-function is a strange beast. It was originally crafted by John Conway as a counterexample to the converse of the intermediate value theorem, and has the property that on any open interval its image contains the entire real line. In addition, its support set also serves as an illustration of a dense, uncountable set of numbers whose Lebesgue measure is 0. Read the rest of this entry »