Derivative of Uniformly Convergent Sequence of Differentiable Functions

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Theorem

Let $J$ be a bounded interval.

Let $\sequence {f_n}$ be a sequence of real functions $f_n: J \to \R$.

Let each of $\sequence {f_n}$ be differentiable on $J$.

Let $\sequence {\map {f_n} {x_0} }$ be convergent for some $x_0 \in J$.

Let the sequence of derivatives $\sequence { {f_n}'}$ converge uniformly on $J$ to a function $g : J \to \R$.


Then $\sequence {f_n}$ converge uniformly on $J$ to a differentiable function $f: J \to \R$ and $D_x f = g$.


Proof


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