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Let $C$ be an infinite connected set in $\mathbb{R}$ and $\{f_n\}$ be a sequence of differentiable functions from $C$ to $\mathbb{R}^k$.

Suppose (i)$f_n'$ coverges uniformly $//$ (ii)There exists $s\in C$ such that $\{f_n(s)\}$ converges.

Then, it can be shown that $f_n\rightarrow f$ pointwise and $f_n'\rightarrow f'$ uniformly to some function $f$.

If $C$ is bounded, then it can be also shown that $f_n\rightarrow f$ uniformly.

However, if $C$ is unbounded, then i cannot prove that $f_n\rightarrow f$ uniformly. Is it true or is there a counterexample to this?

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up vote 5 down vote accepted

Let $f_n(x)=\frac1nx$ on $C=\mathbb R$.

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