1. (2 point) Show that if $f$ is integrable on $[a,b]$, then for any sub-interval $[c,d] \subset [a,b]$, $f$ is integrable on $[c,d]$.

2. (2 point) If $f$ is a continuous non-negative function on $[a,b]$, and $\int_a^b f dx = 0$, then $f(x)=0$ for all $x \in [a,b]$.

3. (3 point) Let $f:[0,1] \to \R$ be given by $$ f(x) = \begin{cases} 0 &\text{if } x = 0 \cr \sin(1/x) &\text{if } x \in (0,1] \end{cases}. $$ And let $\alpha: [0, 1] \to \R$ be given by $$ \alpha(x) = \begin{cases} 0 &\text{if } x = 0 \cr \sum_{n \in \N, 1/n<x} 2^{-n} &\text{if } x \in (0,1] \end{cases}. $$ Prove that $f$ is integrable with respect to $\alpha$ on $[0,1]$. Hint: prove that $\alpha(x)$ is continuous at $x=0$.

4. (3 point) Let $p,q>0$ be positive real numbers, such that $1/p + 1/q = 1$. Prove that, if $f, g$ are bounded real functions on $[a,b]$ that are Riemann integrable, then $$ \int fg dx \leq \left[ \int |f|^p dx \right]^{1/p} \left[ \int |g|^q dx \right]^{1/q} $$ Hint: (a) If $u \geq 0, v \geq 0$, then $$ uv \leq \frac{u^p}{p} + \frac{v^q}{q} $$ If you cannot prove this, you may assume it and proceed (no points taken off). If you want to prove it, you may fix $u$ and let $v$ vary from $0$ to $\infty$, and watch how $\frac{u^p}{p} + \frac{v^q}{q} - uv$ change, and obtain that at the minimum the quantity is still non-negative.

(b) If $f, g$ are non-negative Riemann integrable functions on $[a,b]$, and $$ \int f^p dx = 1, \quad \int g^q(x) dx = 1 $$ Show that $\int fg dx \leq 1$.

Suggested reading:
1. Ross theorem 32.7, if a function $f$ is Riemann integrable on $[a,b]$, then as 'mesh-size' of a partition goes to 0, the gap $U(P, f) - L(P, f)$ tends to 0.

2. There is a 'Lebesgue criterion for Riemann integrability', see here. A weaker version that avoids introducing Lebesgue measure is the following: if $f:[a,b] \to \R$ is bounded and real, and $f$ has countably many discontinuities, then $f$ is Riemann integrable. You can try to prove this using a similar strategy to Theorem 6.10 in Rudin.