The Fundamental Theorem of Calculus is as follows:
$\frac{d}{dx}\int_{a}^{x} f(t)dt = f(x)$
I know that to integrate a function $f(t)$, it must be piece-wise continuous. However, I'm not too sure about the conditions under which I can use the Fundamental Theorem of Calculus.
My questions are:
- If $f(t)$ is only piece-wise continuous, will its antiderivative calculated by $\int_{a}^{x} f(t)dt$ have the same finite amount of discontinuities?
- Does this mean that if we were to differentiate this antiderivative as $\frac{d}{dx}\int_{a}^{x} f(t)dt$, the result would be undefined rather than a number?
- Does the differentiability and the continuity of a function carry over for its antiderivative?
Thanks