Given two continuously differentiable functions $f,\ g:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(x_0) = g(x_0)$ and $f'(x_0) < g'(x_0)$, there exists $\epsilon > 0$ such that $f(x) < g(x)$ for $x \in (x_0, x_0 + \epsilon)$.
The above result is not too difficult to prove, but I was wondering if the condition that the functions be continuously differentiable is absolutely necessary. I have not taken much analysis, but I'm wondering if the fact that the derivative exists at $x_0$ would be enough to prove this result without needing the derivative to be continuous. The reason I ask this is because it seems that the existence of a derivative function already implies it must satisfy some quite stringent requirements (i.e. Darboux's Theorem), perhaps these are enough?