Continuous mappings preserve almost everywhere convergence

Question

Let $f_n:\mathbb{R}^d \rightarrow \mathbb{R}$. Given a sequence of maps $(f_n)$ that converges almost everywhere to $f$ and a continuous map $g:\mathbb{R} \rightarrow \mathbb{R}$, is it true that $g(f_n)\rightarrow g(f)$ almost everywhere? I think this should be true because continuous maps preserve convergent sequences.