A question about Integrability and Uniform Continuity

Question

I got this questions:

Prove or disprove by a counterexample the following statements:

Let $f:\mathbb{R}\to\mathbb{R}$ be a function that is integrable on every closed interval and let $F(x)=\int_0^x f(t)\;dt$. (1) If $f$ is bounded on $\mathbb{R}$, Then $F$ is uniformly continuous on $\mathbb{R}$.
(2) If $f$ is continuous on $\mathbb{R}$, Then $F$ is uniformly continuous on $\mathbb{R}$.

Some hints will be helpful. Thanks.

Answer 1

Hint: Only one of the two statements is true. The other can be disproven using very simple functions.