Prove that integral of continuous function is continuously differentiable

I can show you how to get the derivative, but I don't know about proving continuity, etc. According to the fundamental theorem of calculus, there exists a function $G(x)$ such that:

$$ F(x) = \int\limits_a^{g(x)} f(x)dx = G\left(g(x)\right) - G(a)\text{, where } G'(x) = f(x) \\ F'(x) = g'(x)G'\left(g(x)\right) - 0 = g'(x)f\left(g(x)\right) $$

I don't really like the following notation, but problems like these seem to warrant it...you could also write it like this:

$$ F'(x) = g'(x) \cdot \left(f\circ g\right)(x) $$

Hint: $F = G\circ g$ where $G(x) = \int_0^x f(t)\,dt$ is a function that you can differentiate.