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Suppose $f : \Bbb R → \Bbb R$ is differentiable on $\Bbb R$, and let $(x_n)$ be a Cauchy sequence. Is the sequence $(f′(x_n))$ also Cauchy?

My first thought was to take $f(x)=\log x$, and $(x_n)=1/n$. But $\log x$ is not defined for $x=0$. I am unable to think of any other counterexample. Is $(f′(x_n))$ Cauchy?

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    Does it matter if you take a sequence of positive numbers? – timur Jan 09 '22 at 23:19
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    Given the completeness of $\mathbb{R}$, this is equivalent to asking if $f'$ is continuous. You can find an example of a discontinuous derivative (about as elementary as it gets) at the top of Mark McClure's answer here. – Theo Bendit Jan 09 '22 at 23:19

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