Riemann integral

Question

For a function $f$ to be Riemann integrable on an interval $[a, b]$ does $f$ have to be continuous for all $x \in [a, b]$? Also does this function have to be vertically bounded?

Answer 1

$f$ need not be continuous. For example, $$f(x)=\begin{cases}0&\text{if }x<1\\ 1&\text{if }x\ge 1\end{cases}$$ is Riemann integrable on $[0,2]$.

But if $f$ is not vertically bounded, you can always find Riemann sums that exceed any given bound, no matter how fine you prescribe the partition of $[a,b]$ to be.