Explain carefully what it means to say that a bounded function $f:[0,1] \rightarrow \mathbb{R}$ is Riemann integrable.

Prove that every continuous function $f:[0,1] \rightarrow \mathbb{R}$ is Riemann integrable.

For each of the following functions from $[0,1]$ to $\mathbb{R}$, determine with proof whether or not it is Riemann integrable:

(i) the function $f(x)=x \sin \frac{1}{x}$ for $x \neq 0$, with $f(0)=0$;

(ii) the function $g(x)=\sin \frac{1}{x}$ for $x \neq 0$, with $g(0)=0$.