Is $F(x)=\int_{a}^xf(t)dt$ continuous everywhere if $$f(x)=\begin{cases} e^x, & \text{if }x>0\\ C, & \text{if C is constant not equal 1 and }x\le0 ?\\ \end{cases}$$ This confuse me since I know $f(x)$ is not continuous at $0$ but $F(x)$ is a sum of area which seems always continuous.
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So $F(x)$ is continuous iff $f(x)$ is bounded in [a,x]? – joefu Sep 27 '15 at 10:57
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1This is not what the link says. It says $F$ is continuous if $f$ is Riemann integrable. Which is true for your example on every bounded interval. – Thomas Sep 27 '15 at 11:02
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(and note: I wrote 'if', not 'iff') – Thomas Sep 27 '15 at 11:05