I'm wondering about the following.
First of all $f$ does not need to be continuous in order to be integrable, or? So suppose my function $f$ is not continuous but integrable. Further $f_n$ is a sequence of functions approximating $f$ and $$\lim_{n\rightarrow \infty} f_n = f$$ including the discontinuity (e.g. a jump).
Then still $$ \lim_{n\rightarrow \infty} \int_{a}^{b} f_n = \int_a^b f$$ holds?
EDIT: Let's now suppose $f'(x)$ is discontinuous, then why is $$F(x)=\int_{0}^{1} f(tx)x \, {\rm d}t = \int_0^x f(t) \, {\rm d}t$$ not an antiderivative?
I'm asking, because in the context of holomorphic functions it is (apparently) required that the integrand of $$F'(x)=\int_{0}^{1} \left\{f(tx) + f'(tx)tx\right\} {\rm d}t = \int_0^1 {\rm d}t \, \frac{{\rm d}}{{\rm d}t} \, tf(tx)$$ needs to be continuous.