Problem 1. (a) Let $f(x) = 0$ for $x < 0$ and $f(x) = x$ for $x \ge 0$. Whether $f(x)$ is differentiable at $x = 0$? Please justify your answer. (Hint: $\lim_{x \to a} f(x)$ exists if and only if one-side limits exist and $\lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x)$.)
(b) Consider $g(x) = 0$ for $x < 0$ and $g(x) = x^2$ for $x \ge 0$. Whether $f(x)$ is differentiable at $x = 0$? Please justify your answer.
Problem 2. Suppose that $|f'(x)| \le L$ for all $x \in (a, b)$. Show that $|f(x) - f(y)| \le L|x - y|$ for all $x \in (a, b)$. Conclude that $|\cos x - \cos y| \le |x - y|$ for all $x, y \in \mathbb{R}$.
Problem 3. (Hard) Let $f$ be differentiable on $\mathbb{R}$ with $a = \sup\{|f'(x)| : x \in \mathbb{R}\} < 1$.
(a) Given $s_0 \in \mathbb{R}$ and define $s_n = f(s_{n-1})$ for $n \ge 1$. Thus $s_1 = f(s_0)$, $s_2 = f(s_1)$, etc. Prove the sequence $(s_n)$ is convergent. (Hint: To show that $(s_n)$ is Cauchy, first show $|s_{n+1} - s_n| \le a|s_n - s_{n-1}|$ for $n \ge 1$.)
(b) Prove $f$ has a fixed point, i.e., $f(s) = s$ for some $s$ in $\mathbb{R}$.
