$$
\newcommand{\norm}[1]{\left\#1\right\}
\newcommand{\paren}[1]{\left(#1\right)}
\newcommand{\abs}[1]{\left\lvert#1\right\rvert}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\ang}[1]{\left\langle#1\right\rangle}
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
$$
Due: December 8th, 2017
Math 142A Assignment 8
Answers to these problems are now available here.
The $\LaTeX$ file used to produce them is here.

Suppose that $f : [a, b] \to \mathbb{R}$ is continuous, $x_0 \in (a, b)$, and $f$ is differentiable on $(a, b) \setminus\{x_0\}$.
Further suppose that $\displaystyle\lim_{x\to x_0} f'(x)$ exists and equals $L$.
Show that $f$ is differentiable at $x_0$, and $f'(x_0) = L$.

Consider the function
$$\begin{align}f : \R &\longrightarrow \R\\ x&\longmapsto \begin{cases}x + x^2\sin\paren{\frac1{x^2}} & \color{red}{\text{ if } x \neq 0} \\ \color{red}{0} & \color{red}{\text{ if } x = 0}.\end{cases}\end{align}$$
 Confirm that $f$ is differentiable at $0$, and $f'(0) = 1$.
 Prove that there is no interval of the form $[\delta, \delta]$ with $\delta > 0$ on which $f$ is increasing.
(In fact, $g$ defined by $g(x) = x+x^2\sin\paren{\frac1x}$ also has this property, but it is a little harder to demonstrate it.)

 Show that if $f$ is continuous on $[a,b]$ but not monotonic, there are points $a \leq x_1 \le x_2 \leq b$ so that $f(x_1) = f(x_2)$.
 Suppose that $f : [a\epsilon, b+\epsilon] \to \R$ is differentiable on $[a,b]$, and $f'(a) \color{red}\lt 0 \color{red}\lt f'(b)$. Show that $f$ is not strictly monotonic on $[a,b]$.
 Imagine that $f : [a\epsilon, b+\epsilon] \to \R$ is differentiable on $[a,b]$ and $f'(a) \color{red}\lt M \color{red}\lt f'(b)$. Show that there is some $c \in (a,b)$ so that $f'(c) = M$. (You may wish to consider the function $x \mapsto f(x)  Mx$.)
 Show that there is no function $f : \R\to\R$ so that
$$f'(x) = \begin{cases}1&\text{ if } x \in \Q \\ 0 & \text{ if } x \notin \Q.\end{cases}$$

Bonus:
Show that parts B and C become false if the red inequalities above are $\leq$ rather than $\lt$.

Suppose that $f : (1, 1) \to \R$ has $n$ derivatives on $(1,1)$.
 Show that if the $n$th derivative $f^{(n)}$ is bounded and $$f(0) = f'(0) = \cdots = f^{(n1)}(0) = 0,$$ then $f(x) \leq Mx^n$ for some $M > 0$ and all $x \in (1,1)$.
 Show that if $f(x) \leq Mx^n$ for some $M > 0$ then $$f(0) = f'(0) = \cdots = f^{(n1)}(0) = 0.$$

Show that if $f : (a,b)\to \mathbb{R}$ and $c \in (a,b)$, there is at most one function $F : (a, b) \to\R$ so that $F(c) = 0$ and $F'(x) = f(x)$.

Bonus:
Does there exist a function $f : \R \to \R$ which is smooth (i.e., infinitely differentiable at every point in its domain) and so that all the derivatives of $f$ vanish at $0$ (i.e., $f^{(n)}(0) = 0$ for all $n \geq 0$), but $f(1) \neq 0$?