= Uniform derivative convergence with an anchored value
If continuously differentiable functions have uniformly convergent <derivatives> on an interval and their values converge at one point, the functions converge locally uniformly to a continuously differentiable limit whose <derivative> is the <derivative> limit. The proof integrates from the anchored point using the <fundamental theorem of calculus>. <Uniform convergence> of the functions on the entire interval follows when the interval is bounded; it need not follow on an unbounded interval.
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