Due: November 8th, 2017

Math 142A Assignment 5

Answers to these problems are now available here. The $\LaTeX$ file used to produce them is here.

1. Prove that, at any given instant, there are two antipodal points on the equator with the same temperature. (You may assume that temperature varies continuously with position, and that the equator is a perfect circle; antipodal points are points on either end of a diameter of the circle.)
2. Let $h : \mathbb{R} \to \mathbb{R}$ be defined so that $h|_{[0, 1]} : x \mapsto x$, $h|_{[1, 2]} : x \mapsto 2-x$, and for all $x \in \mathbb{R}$, $h(x) = h(x+2)$. Note that $h$ behaves as a triangle wave, and that $h(x) = h(-x)$.

Define a function $f$ as follows: \begin{align*}f : \mathbb{R} &\longrightarrow \mathbb{R} \\ t &\longmapsto \begin{cases} h\left(\frac1x\right) & \text{ if } x \neq 0 \\ 0 & \text{ if } x = 0. \end{cases}\end{align*}

1. Show that $f$ is discontinuous at $0$, and continuous everywhere else.
2. Show that $f$ satisfies the conclusion of the intermediate value theorem: for every closed interval $[a, b]$ with $a \neq b$, and every $c$ between $f(a)$ and $f(b)$, there is some $x \in (a, b)$ so that $f(x) = c$.

What we've shown here is that not every function which satisfies the conclusion of the intermediate value theorem is continuous. In fact, things can be much, much worse: the Conway base 13 function takes on every real value in every non-empty open interval — in particular, it has the intermediate value property — but is horribly discontinuous at every point.

3. Show that if $f, g : X \to \mathbb{R}$ are continuous functions, then so is $f \vee g : t \mapsto \max(f(t), g(t))$. Show that the analogous statement about uniformly continuous functions is true.
4. Suppose $S \subseteq \mathbb{R}$ is not sequentially compact. Show that there is a continuous positive function $f : S \to (0, \infty)$ which does not achieve its minimum value. (Hint: you may wish to treat separately the case that $S$ is unbounded and the case that $S$ is not closed.)
5. Suppose that $f : (a, b) \to \mathbb{R}$ is a uniformly continuous function on the open bounded interval $(a, b)$. Show that $f$ is bounded.
6. Bonus:

The supremum norm of a function $f : X \to \mathbb{R}$ is defined to be the quantity $$\|f\|_\infty = \sup\{|f(x)| : x \in X\}$$ if the supremum exists, and infinity otherwise. A sequence of functions $(f_n)$, with $f_n : X \to \mathbb{R}$, is said to converge uniformly to a function $f : X \to \mathbb{R}$ if for every $\epsilon > 0$ there is $N \in \mathbb{N}$ such that for all $n > N$, $$\|f_n - f\|_\infty < \epsilon.$$ The sequence is said to converge to $f$ pointwise if for every $x \in X$, the sequence of real numbers $(f_n(x))$ converges to $f(x)$. Observe that uniform convergence implies pointwise convergence.

1. Prove that if $(f_n)$ is a sequence of continuous functions which converges uniformly to $f$, then $f$ is continuous.
2. Let $g_n : [0, 1] \to \mathbb{R}$ be defined by $g_n : x \mapsto x^n$. Show that $g_n$ converges pointwise to a discontinuous function.
3. Much more difficult:

Suppose that $(f_n)$ are a sequence of functions with $f_n : [a, b] \to [-M, M]$ such that for every $\epsilon > 0$ there is a $\delta > 0$ such that for every $n \in \mathbb{N}$ and $x, y \in [a, b]$, if $|x-y| < \delta$ then $|f_n(x) - f_n(y)| < \epsilon$. Show that there is a subsequence $(f_{n_k})$ which converges uniformly to some $f : [a, b] \to [-M, M]$.
7. (You don't need to turn this one in, but think about it for a couple minutes.) Does there exist an unbounded set $X \subseteq \mathbb{R}$ such that every continuous function $f : X \to \mathbb{R}$ is uniformly continuous?