Problem 1 and 2 involves Riemann integral $\int f dx$ instead of the more general Riemann Stieltjes integral.

1. (2 point) Show that if $f$ is integrable on $[a,b]$, then for any sub-interval $[c,d] \In [a,b]$, $f$ 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 \\
 \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 \\
 \sum_{n \in \N, 1/n<x} 1/2^n &\text{if } x \in (0,1]

\end{cases}. $$ Is $f$ integrable with respect to $\alpha$ on $[0,1]$?

4. (3 point)