Bounding a function by taking an integral of a bound on its derivative, if the derivative is not necessarily integrable

Question

Let $f$ be real-valued continuous function on $[a,b]$, differentiable on $(a,b)$.

Let $g$ be Riemann integrable on [a,b].

Assume that $f'(x) \geq g(x)$ when $x \in (a,b)$

Does this imply that $f(x) \geq f(a) + \int_a^x g(t) dt$ for $x\in [a,b]$ ?

Comment: It works if $f'$ is Riemann integrable.