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Let $$f(x)=\begin{cases}\exp\left(1-\frac{1}{1-|x|}\right),\,\,&|x|<1\\0,&\text{otherwise}.\end{cases}$$ and show that $f$ is continuously differentiable.

I took the derivative and obtained $$f'(x)=\begin{cases}\frac{\text{sgn}(x)}{(1-|x|)^2}\exp\left(1-\frac{1}{1-|x|}\right),\,\,&|x|<1\\0,&\text{otherwise}.\end{cases}$$ which does not look to be continous at $x=0$ and thus is not continuously differentiable. Did I do something wrong?

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From what you've written here, I can't see anything wrong, except a misssing negative sign in front of everything. The only thing I can think of is that you might have copied it wrong. If not, then the problem itself is wrong (presumably there is a typo somewhere).

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