1. (2 pt) One corollary of the intermediate value theorem for derivative is the following (Rudin page 109): If $f$ is differentiable on $[a,b]$, then $f'$ cannot have any simple discontinuities on $[a,b]$. Give a proof of this statement.

2. (2 pt) Let $f_n(x)$ be a sequence of differentiable functions on $[0,1]$, convergent uniformly to $f(x)$. Is it true that $f(x)$ is differentiable? If not, give an example; if true, give a proof.

3. (3 pt) Let $f(x) = x^4(2 + \sin(1/x))$ for $x \neq 0$ and $f(0)=0$. Compute its derivative and prove that there is a sequence of non-zero local minimum convergent to $0$.

Hint: (1) what are the local min and local max for $2 + \sin(1/x)$? (2) How does multipling the factor $x^4$ change your previous answer? (3) You can try to change the variable, let $u=1/x$, then the function become $(2 + \sin(u)) / u^4$, the question then becomes: “can you find a sequence of local minimums as $u$ goes to $+\infty$?”.

click here for a log scale plot. Ploted using SageMath

If you find constructing local minimum too difficult, you can prove something weaker: there is a sequence of non-zero critical points of $f$ convergent to $0$, where a critical point of $f$ is a point $x$ with $f'(x)=0$.

4. (3 pt) Let $\varphi(x) = \min \{ |x - n| | n \in \Z \}$, then $\varphi(x)$ is a periodic continuous function, with a shape of saw-teeth. Plot $\varphi(x)$. We will use $\varphi(x)$ to construct a continuous and nowhere differentiable function. Prove that (updated version, replaced $2^n$ by $4^n$) $f(x) = \sum_{n=0}^\infty 4^{-n} \varphi(4^n x)$ is such a function.

Hint: Let $\varphi_n(x) = 4^{-n} \varphi(4^n x)$. For each point $x \in \R$ and each positive integer $n$, show that we have $h_n = 4^{-n-1}$ or $h_n=-4^{-n-1}$ such that that $|\varphi_n(x + h_n) - \varphi_n(x)| = |h_n|$. Then show that, for any integer $m > n$, then $\varphi_m(x+h_n) = \varphi_m(x)$, and for any $m < n$ we still have $|\varphi_m(x + h_n) - \varphi_m(x)| = |h_n|$. Then, the quotient $(f(x+h_n) - f(x))/h_n$ is a sum of $n+1$ terms of $\pm 1$, and you may deduce some contradiction to $\lim_{n \to \infty} (f(x+h_n) - f(x))/h_n$ exist.

Extra question (you are free to take a guess. This question has no points). Let $f(x)$ be a differentiable function on $[-1,1]$ with $f'(x)$ continuous. Assume $x=0$ is the unique global minimum of $f$, i.e., for any $x \neq 0$, we have $f(x) > f(0)$. Is it true that there exists a $\delta > 0$, such that $f'(x) < 0$ for $x \in (-\delta, 0)$ and $f'(x)>0$ for $x \in (0 ,\delta)$? (Just as the case if $f(x)=x^2$)?