Let $f$ be Lebesgue integrable on the real line. For $\chi_A$ (the characteristic function of the set $A$), we know that $\int_Ef=\int_{\chi_E}$. Then, define:
$$F = \begin{cases} \int_{[0,x]}f &\text{if } x\ge 0 \\ \ - \int_{[0,x]}f &\text{if } x\lt 0 \end{cases} $$
Show that:
1) $F$ is continuous, and
2) If $F = 0$, then $f \stackrel{a.e.}{=} 0$.
I know that I need to find sequences (fn) that converge and that their left and right limits are equal. I think I need to use the DMC, but I have no idea on how exactly to apply it and find these limits.
2) requires Lebesgue's Theorem which says that $\frac 1 h\int_x^{x+h} f(y)\, dy \to f(x)$ as $ h \to 0$ for almost all $x$. Since $\int_x^{x+h} f(y)\, dy=F(x+h)-F(x)$ we get 2) immediately.
1) follows from the fact that $|\int_A f(x)\, dx| \to 0$ as $m(A) \to 0$. [ $m$ being Lebesgue measure]. This statement can be verified easily when $f$ is a simple function. For the general case use the fact that given $\epsilon >0$ there is a simple integrable function $g$ such that $|\int f-\int g |<\epsilon$.