Uniform convergence and differentiation

Uniform convergence and differentiation: Let $f_n:[a,b]\to\mathbb{R}$ be differentiable on $[a,b]$ (or on $(a,b)$ with suitable endpoint control). Assume:

• there exists $x_0\in[a,b]$ such that the sequence $(f_n(x_0))$ converges in $\mathbb{R}$, and
• the derivatives $f_n'$ converge uniformly on $[a,b]$ to a function $g$.

Then $f_n$ converges uniformly on $[a,b]$ to a differentiable function $f$, and $f'(x)=g(x)\quad \text{for all } x\in[a,b].$

This theorem provides the rigorous justification for differentiating a sequence (or series) term-by-term, under uniform convergence of derivatives.