Prove that the function $f$ pointwise limit of a function sequence $(f_n)$ is continous, where $(f_n)$ follows a certain property.

Question

Let $f_n: I \to \Bbb{R}$ be a sequence of functions such that for all $(x_n)$ a sequence that conveges, the sequence $(f_n(x_n))$ converges. ($I$ is an interval)

$(f_n)$ converges pointwise to a function $f$ (take a constant sequence). I want to prove that $f$ is continous.

Any tips?

Answer 1

There is a rule that if interval $I$ is closed and finite, then $f$ is continuous and monotonous, with terms $\min$ and $\sup$