The function $f : \mathbb{R}\rightarrow \mathbb{R}$ is defined by
$$f(x)= \left\{\begin{matrix}
e^{-\frac{1}{x}}, & x > 0\\
0,& x \leq 0\;.
\end{matrix}\right.$$
Then
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$f$ is not continuous
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$f$ is continuous, but not differentiable everywhere
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$f$ is differentiable but $f’$ is not continuous
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$f$ is differentiable and $f’$ is continuous