Let $\{f_n\}$ be a sequence of continuous functions $f_n\colon X \to R$ where $X \subset R$. Prove that if every point $x_0 \in X$ has an interval $ I_{x_0}=B\left({x_0 ,\varepsilon _{x_0}}\right)\cap X$ for some $\varepsilon _{x_0 } >0$, such that $\{f_n\}$ converges uniformly in $I_{x_0 } \cap X$, then $\{f_n\}$ it´s also equicontinuous.
Is this result true for a non-countable set of continuous functions? Is the reciprocal true? Is this result true for other kind of topological, or metric spaces? Please help me with this, to start, it's the only that i could not do.