Continuous_Limit_Theorem

Theorem

Let $\left. \left( f_{n} \right) \right.$ be a sequence of functions defined on $A \subseteq \mathbb{R}$ that converges uniformly on $A$ to a function $f$. If each $f_{n}$ is continuous at $c \in A$, then $f$ is continuous at $c$.

Proof: See page 179 of Understanding_Analysis.