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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?

user2850514
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    No to the first question. See this. If by "vertically bounded" you mean "bounded" and if "this function" is Riemann integrable, then the answer to your second question is "yes". – David Mitra Feb 02 '14 at 15:29

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$f$ 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.

Hagen von Eitzen
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