Pointwise convergence of a sequence of continuous monotone functions to a continuous function is uniform?

Question

Let $f_n$ be a sequence of continuously differentiable real-valued functions converging pointwise to a continuous function $f$. Assume further that each of the $f_n$, and $f$ itself, is a strictly monotone increasing function (note: the sequence is not necessarily monotone in $n$).

Does this imply that $f_n$ uniformly converges to $f$? And that $f_n$ is therefore equicontinuous?

Answer 1

This is false on $[0,1).$ Consider $f_n(x) = x + x^n,f(x) = x.$