Let $f:\mathbb{R} \mapsto [0,+\infty]$ , $\lambda$ the Lebesgue measure in $\mathbb{R}$ and $f\in L^1(\lambda)$. Show that the function $F(x)=\int \limits_{-\infty}^xf(t)dt$ is continuous.
I tried to use the $\epsilon- \delta$ definition of continuity but I am not sure that will work, or how it will work. Any ideas?
Let $(x_n)_{n\in\mathbb N}$ be a sequence of real numbers converging to $x_0$. For each $n\in\mathbb N$, let$$f_n(x)=\begin{cases}f(x)&\text{ if }x\leqslant x_n\\0&\text{ otherwise.}\end{cases}$$Then $(f_n)_{n\in\mathbb N}$ converges almost everywhere to $f(x)$ if $x<x_0$ and to $0$ otherwise. It follows from Lebesgue's dominated convergence theorem that$$\lim_{n\to\infty}F(x_n)=F(x).$$