Can you find two functions $f$ and $g$ defined on a closed interval $[a, b]$, with real values, such that $\exists (x_n) $ an infinite sequence of distinct points in $[a, b] $ such that $$\forall n, f(x_n) =g(x_n) $$ but $$f\neq g$$
EDIT: the question is not interesting as it is stated. I therefore require that $f$ or $g$ do not coincide on any interval.
I have an answer when they are defined on $\mathbb R$, but in this case, I can't find such.
Bonus point if you find some that are continuous, or $C^n$ for large $n$