Uniform limit theorem for continuity

Uniform limit theorem for continuity: Let $(X,d)$ be a metric space and let $(f_n)$ be a sequence of continuous functions $f_n:X\to\mathbb{R}$ (or into any metric space). If $f_n\to f$ uniformly on $X$, then $f$ is continuous on $X$.

Uniform convergence is strong enough to pass continuity through the limit. This is a fundamental reason uniform convergence is preferred over pointwise convergence in analysis.