If $f$ is invertible in $A \subset M$, $\lim_n \sum_{x \in f^{-1}(y)} g_n (y) = \chi_{f(A)}(y)$?

Question

If $f$ is invertible in $A \subset M$, $M$ compact metric space, and $(g_n)_n$ is a sequence of continuous functions converging to $\chi_A$, with $\sup|g_n| \leq \sup \chi_A = 1 $ then

$\lim_n \sum_{x \in f^{-1}(y)} g_n (y) = \chi_{f(A)}(y)$.

I could not do it, any suggestions? I think it is necessary that $f(A) \subset A$.