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 $E : (a, b) \to \R$ continuous at $x$ and a constant $C \in \R$ so that $E(x) = 0$ and for all $t \in (a, b)$, \[f(t) = f(x) + C(t-x) + E(t)(t-x).\] Moreover, show that in this case $C = f'(x)$.

2. 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 $x \in \R$.

3. More derivatives

   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 $0$.

4. Inverses

   Suppose that $f : (a, b) \to \R$ is differentiable with $f'(x) \gt 0$ for all $x \in (a, b)$.

   1. Prove that $f$ is strictly monotonically increasing, i.e., if $a \lt x \lt y \lt b$ then $f(x) \lt f(y)$.
   2. Note that $f$ is one-to-one, and so has an inverse function $g : f((a, b)) \to (a, b)$. Show that $g$ is continuous.
   3. Show that for all $x \in (a, b)$, then we have $f(x) \in f((a, b))^\circ$ (the interior of $f((a,b))$). (Fixed grammar.)
   4. We saw that if $g$ is differentiable at $f(x)$, it must be the case that \[g'(f(x)) = \frac1{f'(x)}.\] Verify that $g$ is differentiable at $f(x)$.

5. Positive derivative at a point

   Suppose that $f : \R \to \R$ is differentiable, and $f'(0) \gt 0$.

   1. Show that if $f'$ is continuous at $0$, then there is some $\delta\gt0$ so that $f$ is monotonically increasing on $(-\delta, \delta)$.
   2. Show by example that this need not be the case if $f'$ is not continuous at $0$.

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.

• Differentiability and monotony

  1. Does there exist a differentiable function $f : \R\to\R$ which is not strictly monotonic on any open interval?
  2. Does there exist a differentiable function $f : \R\to\R$ which is not monotonic on any open interval? (Recall that monotonic differs from strictly monotonic in that the strict inequalities $\lt$, $\gt$ in its definition are replaced by weak inequalities $\leq$, $\geq$.)
  3. Does there exist a differentiable function $f : \R \to \R$ so that $f'(0) = 0$ but $f$ is strictly monotonic?

• 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 8 Question 4?

  (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".))