Uniform limit of continuous functions is continuous

Uniform limit of continuous functions is continuous: Let $X$ be a metric space and let $Y$ be a metric space. If each $f_n:X\to Y$ is continuous and $f_n\to f$ uniformly on $X$, then $f:X\to Y$ is continuous.

This is one of the key ways uniform convergence improves on pointwise convergence : continuity is preserved under uniform limits but not under pointwise limits in general.