= Solution
<Continuity> at $x_0$ requires that $f(x_0)$ be defined and that $\lim_{x\to x_0}f(x)=f(x_0)$. In the epsilon-delta definition, for each $\varepsilon>0$ there must be a $\delta>0$ such that $|f(x)-f(x_0)|<\varepsilon$ whenever $|x-x_0|<\delta$ within the domain. <Differentiability> requires the finite two-sided limit
$$
f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}h.
$$
In particular, merely having a punctured <limit of a function> does not imply <continuity> unless the assigned value agrees with that limit.
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