There is a theorem saying that if $f:[a,b]\to \mathbb R$ is integrable on $[a,b]$, then $F(x):=\int_{a}^{x}f(t)dt$ is continuous on $[a,b], x \in [a,b]$.
Is there an analogous theorem of the kind: if $f:\mathbb R \to \mathbb R$ and $F(x):=\int_{-\infty}^{\infty}f(t)dt$ converges, then $F(x):=\int_{-\infty}^{x}f(t)dt$ is continuous on $\mathbb R, x \in \mathbb R$?
If the theorem exists, can somebody give a reference or a sketch of the proof?