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Sequence of continuously differentiable functions

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Q: Prove that if {f_n:R→R} is a sequence of continuously differentiable functions such that the sequence of derivatives {f_n^':R→R} is uniformly convergent and the sequence {f_n (0)} is also convergent, then {f_n:R→R} is pointwise convergent. Is the assumption that the sequence {f_n (0)} converges necessary?

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Mathematics: Sequence of continuously differentiable functions

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