The Antiderivative is uniformly continuous if the function is continuously differentiable

Let $g$ be a continuously differentiable function on $\mathbb{R}$. Then, is the function $\int_0^xg\ dt$ uniformly continuous on $\mathbb{R}$? Is it also lipschitz continuous? I think it is uniformly continuous, by using the fundamental theorem of calculus(or mean value theorem), but am unsure about the rigour of proof required. Any hints. Thanks beforehand.

Neither. $F(x)=x^2$ - or any polynomial of degree $\ge2$, for that matter - is not UC on $\Bbb R$, so $g(x)=2x$ is a counterexample.