1. Let $f: \R \to \R$ be a differentiable function. Assume that $f$ is convex, namely for any $x, y \in \R$ and any $t \in [0,1]$, we have $$ tf(x) + (1-t) f(y) \geq f(t x + (1-t) y).$$ Prove that $f'(x)$ is monotonously increasing.

2. Let $f: (0, \infty) \to \R$ be a twice differentiable function. Suppose $f$'' is bounded, and $f(x) \to 0$ as $x \to \infty$. Show that $f'(x) \to 0$ as $x \to \infty$.

Hint: Use Taylor Theorem, prove that for any $x>0, h>0$, we have $$ \frac{f(x+h) - f(x)}{h} = f'(x) + (h/2) f“(\xi) $$ for some $\xi \in (x, x+h)$.

3. Let $f: \R \to \R$ be a continuous function, such that $f'(x)$ exists for all $x \neq 0$. If we know that $\lim_{x \to 0} f'(x)=5$, can we conclude that $f'(0)=5$? Justify your result.

4. Given an example of real bounded function $f$ on $[0,1]$ that is not Riemann integrable. Hint: $f$ is not continuous on $[0,1]$. Show that $U(f) \neq L(f)$.

5. (Free discussion problem, no points taken). Let $f: [0,1] \to \R$ be a real bounded function. Assume $f(x) \geq 0$ and $f$ is Riemann integrable. Here is a candidate that measures the area under the graph of $f$:

• For each positive integer $n$, let $N(n)$ be the number of points $(x,y) \in \R^2$, such that $x, y \in \Z / 2^n$, and $x \in [0,1], 0 \leq y \leq f(x)$. Namely, we count how many points in the 2d mesh with spacing $2^{-n}$ falls within the area under the curve $y=f(x)$.
• Let $A(n) = 2^{-2n} N(n)$. Since the area of a $2^{-n} \times 2^{-n}$ square is $2^{-2n}$.

Is it true that $$ \lim_{n \to \infty} A(n) = \int_0^1 f(x)dx ? $$ To get started, you may assume that $f$ is continuous (we will see next week that continuous functions are Riemann integrable).