A small part of a chain rule proof I don't get

Hello,

I've been trying to get my head round part of a proof and I don't see the following implication.

Suppose we have

$\displaystyle \lim_{x \to x_0} E(g(x)) = E(g(x_0)) = 0$

then does it follow that

$\displaystyle \lim_{x \to x_0}[f'(g(x_0) + E(g(x)))]\frac{g(x) - g(x_0)}{x - x_0} = f'(g(x_0))g'(x_0)$?

Since doesn't the above require that $\displaystyle f'$ is continuous at $\displaystyle g(x_0)$, and if it does, why is this necessarily true?

Thanks :)

Stonehambey