Explain what it means for a function $f:[a, b] \rightarrow \mathbb{R}$ to be Riemann integrable on $[a, b]$, and give an example of a bounded function that is not Riemann integrable.

Show each of the following statements is true for continuous functions $f$, but false for general Riemann integrable functions $f$.

(i) If $f:[a, b] \rightarrow \mathbb{R}$ is such that $f(t) \geq 0$ for all $t$ in $[a, b]$ and $\int_{a}^{b} f(t) d t=0$, then $f(t)=0$ for all $t$ in $[a, b]$.

(ii) $\int_{a}^{t} f(x) d x$ is differentiable and $\frac{d}{d t} \int_{a}^{t} f(x) d x=f(t)$.