Math 104 Assignment 10

1. A criterion for differentiability

   Suppose $f : (a, b) \to \R$ and $x \in (a, b)$. Show that $f$ is differentiable at $x$ if and only if there is a function $h : (a, b) \to \R$ and a constant $C \in \R$ so that \[\lim_{t\to x}\frac{h(t)}{t-x} = 0 = h({\color{red}x})\] and for all $t \in (a, b)$, \[f(t) = f(x) + C(t-x) + h(t).\] Moreover, show that in this case $C = f'(x)$.

   (Another equivalent condition is that there is a function $E : (a, b) \to \R$ and a constant $C \in \R$ so that $E(x) = 0$, $E$ is continuous at $x$, and \[f(t) = f(x) + C(t-x) + E(t)(t-x).\] It's probably worth thinking about why this is also equivalent, but you don't need to prove it.)

2. Path connected sets

   Suppose $M$ is a metric space. If $x, y \in M$, a path (in $M$) from $x$ to $y$ is a continuous function $\gamma : [0, 1] \to M$ with $\gamma(0) = x$ and $\gamma(1) = y$.

   Define a relation $\sim_p$ on $M$ by $x \sim_p y$ if and only if there is a path from $x$ to $y$ in $M$. As before, set \[ [x]_{\sim_p} = \set{y \in M | x \sim_p y}.\]

   1. Show that $\sim_p$ is an equivalence relation: that is,
      1. for any $x \in M$, $x \sim_p x$;
      2. for any $x, y \in M$, if $x \sim_p y$ then $y \sim_p x$; and
      3. for any $x, y, z \in M$, if $x\sim_p y$ and $y \sim_p z$ then $x \sim_p z$.

   The equivalence classes $[x]_{\sim_p} \subseteq M$ are called path components of $M$. If $M$ has exactly one path component, it is called path connected. (Note that $\emptyset$ is not path connected: it has zero path components, not one.)

   2. Show that if $M$ is path connected, then it is connected.

   3. Suppose that $E \subseteq \R^n$ is open. Show that the path components of $E$ are open (in $\R^n$).

      (If you find it useful, you may use without proof the fact that functions of the form \begin{align*}f : \R &\longrightarrow \R^n \\ t &\longmapsto \vec{a} + t\vec{b}\end{align*} are continuous, where $\vec{a}, \vec{b} \in \R^n$.)

   4. Prove that if $E \subseteq \R^n$ is open and connected, then $E$ is path connected.

   It is not true that connected sets are path connected in general. For example, the set \[T = \set{(0, y) \mid -1 \leq y \leq 1} \cup \set{\paren{x, \sin\paren{\frac1x}} \mid x \gt 0} \subseteq \R^2\] can be shown to be connected but not path connected; its path components are the two separate pieces in the presentation above.

3. Some counterexamples of converses

   Give an example of continuous functions $f, g : \R \to \R$ so that:

   1. $f+g$ is differentiable but $f$ is not.
   2. $fg$ is differentiable, $f$ is not, and $g(x) \gt 0$ for all $x \in \R$.

4. A discontinuous derivative

   Let $S, C : \R \to [-1, 1]$ be differentiable functions with the following properties:

   • for all $x \in \R$, $S(x+\pi) = -S(x)$ and $C(x+\pi) = -C(x)$;
   • for all $x \in \R$, $S'(x) = C(x)$ and $C'(x) = -S(x)$; and
   • $C(0) = 1 = S\paren{\frac\pi2}$ while $C\paren{\frac\pi2} = 0 = S(0)$.

   Let \begin{align*} f : \R &\longrightarrow \R \\ t &\longmapsto \begin{cases} 0 & \text{ if } t = 0 \\ t^2S\paren{\frac1t} & \text{ if } t \neq 0.\end{cases} \end{align*}

   1. Show that $f$ is differentiable on $\R$, and find its derivative (in terms of $C$ and $S$).
   2. Show that $f'$ is discontinuous at $0$.

   (Although we have not finished the proofs yet, you may assume that the Chain Rule holds, and that $h : t \mapsto \frac1t$ is differentiable on $\R\setminus\set0$ with derivative $h'(t) = \frac{-1}{t^2}$.)

5. Limits superior

   Suppose $(a_n)_n$ is a sequence in $\R_\infty$. We define the limit superior of $(a_n)_n$ to be the quantity \[\limsup_{n\to\infty} a_n = \inf \set{ \sup\set{ a_k \mid k \gt n } \mid n \in \N } \in \R_\infty.\]

   1. Let $(b_n)_n$ be the sequence defined by $b_n = {\color{red}\sup}\set{a_k \mid k \gt n }.$ Prove that \[\lim_{n\to\infty} b_n = \limsup_{n\to\infty} a_n.\] (That is, if $\limsup_{n\to\infty} a_n \in \R$, then the sequence $(b_n)_n$ converges to it; otherwise, the sequence $(b_n)_n$ diverges to infinity or to negative infinity according to $\limsup_{n\to\infty} a_n$.)

   2. (Due to an error in its statement, this problem will not be graded.)

      Suppose $t \in \R_\infty$. Show that $t \gt \limsup_{n\to \infty} a_n$ if and only if there are finitely many $n \in \N$ for which $t \lt a_n$.

   3. (Due to an error in its statement, this problem will not be graded.)

      Suppose $t \in \R_\infty$. Show that $t \lt \limsup_{n\to \infty} a_n$ if and only if there are infinitely many $n \in \N$ for which $t \lt a_n$.

   4. Suppose $(a_n)_n$ and $(q_n)_n$ are sequences in $\R$ with finite limits superior. Prove that \[\limsup_{n\to\infty} (a_n + q_n) \leq \paren{\limsup_{n\to\infty} a_n} + \paren{\limsup_{n\to\infty} q_n}.\] Show by example that this inequality may be strict.

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• Limits superior

  1. If you wrote a solution to 5B/5C without noticing the error in the problem, go back and figure out where the error is in your proof.
  2. Prove that if there are finitely many $n \in \N$ for which $t \lt a_n$, then $t \geq \limsup_{n\to\infty} a_n$.
  3. Prove that if there are infinitely many $n \in \N$ for which $t \lt a_n$, then $t \leq \limsup_{n\to\infty} a_n$.
  4. Prove that either of these may happen for $\limsup_{n\to\infty}a_n$ itself.

• A derivative

  Let \begin{align*}f : \R &\longrightarrow \R \\ x &\longmapsto \begin{cases}0 & \text{ if } x \in \Q \\ x^2 & \text{ otherwise}.\end{cases}\end{align*} Show that $f$ is differentiable at $0$, and find its derivative there. (Notice that $f$ is discontinuous everywhere except at $0$! (That is an exclaimation point, not a factorial; $f$ is discontinuous at $1$.))

• Constants

  1. Prove that if $f : \R \to \R$ satisfies $f'(x) = 0$ for all $x \in \R$, then $f$ is constant. (There is actually something to check here.)
  2. Conclude that if $f' = g'$ then $f = g + C$ for some $C \in \R$.

• Growth

  Suppose that $f, g : \R \to \R$ are such that $f(0) = g(0)$ and $f'(x) \lt g'(x)$ for all $x \in \R$. Prove that $\abs{f(x)} \leq \abs{g(x)}$ for all $x \in \R$, with equality only when $x = 0$.

• Difference quotients

  Suppose $f : \R \to \R$. Define a function \begin{align*} F : \R^2 \setminus \set{(x, x) \mid x \in \R} &\longrightarrow \R \\ (x, y) &\longmapsto \frac{f(x) - f(y)}{x-y}. \end{align*} Under what conditions does $F$ extend continuously to $\R^2$, in the sense of Assignment 9 Question 1?

  (While this certainly seems to have something to do with differentiability, it is a little subtle. Notice, for example, that taking a derivative corresponds to holding one variable fixed and letting the other tend to it, or approaching the missing diagonal horizontally or vertically only; meanwhile continuity requires everything in an open ball to be close to the correct value, not just along the two axes. (It just now occurs to me how "axis" and "axe" both have the plural "axes".))

• Intermediate values

  Let us say that a function $f : \R \to \R$ has the "intermediate value property" if and only if for any $a \lt b$ in $\R$, and any $y$ between $f(a)$ and $f(b)$, there is $c \in (a, b)$ so that $f(c) = y$. The Intermediate Value Theorem tells us that continuous functions have the intermediate value property.

  1. Prove that there are discontinuous functions which have the intermediate value property. (In fact, it's possible to find a function $\R \to \R$ which takes on every real value in every non-empty open interval.)
  2. Prove that if $f : \R\to\R$ is differentiable, then $f': \R \to \R$ has the intermediate value property.