$\int_{- \infty}^{+ \infty} |f(t)| dt < \infty \implies \int_{-\infty}^{x} f(t) dt$ is continuous?

Question

I've found counter example for $(A),(D)$ and have shown except a bounded interval $F$ is uniformly continuous everywhere else. And so $(B)$ would imply $(C)$ is correct. But I can't show $(B)$ is correct. Please help someone.

Answer 1

HINT: For (B) consider $G_a(x)=\int_a^x f(t)\,dt$ and remind fundamental theorem of calculus. $F(x)=\int_{-\infty}^a f(t)\,dt + G_a(x)$.