Uniform convergence and differentiation

Uniform convergence and differentiation: Let $f_n:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$ for every $n$. Assume that the sequence of derivatives $f_n'$ converges uniformly on $(a,b)$ to a function $g$, and that there exists $x_0\in[a,b]$ such that the real sequence $(f_n(x_0))$ converges. Then $f_n$ converges uniformly on $[a,b]$ to a function $f$, the limit $f$ is differentiable on $(a,b)$, and

This provides a standard criterion for passing a limit through the derivative operator, complementing results like differentiability implies continuity and uniform limits preserve continuity .