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 local minimum convergent to $0$.

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 $$ f(x) = \sum_{n=0}^\infty 2^{-n} \varphi(2^n x) $$ is such a function.

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 $f(x)=x^2$)?