= Solution
For statement (1), <continuity> at the interior point means
$$
\boxed{\lim_{x\to x_0}f(x)=f(x_0).}
$$
Explicitly, for every $\varepsilon>0$ there exists $\delta>0$ such that $|x-x_0|<\delta$ implies $|f(x)-f(x_0)|<\varepsilon$ for points in the domain. This is the <epsilon-delta definition of continuity>.
For statement (2), <differentiability> means the two-sided difference quotient has a finite <limit>:
$$
\boxed{f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}h\in\mathbb R.}
$$
The same finite value must be obtained from positive and negative $h$. <Differentiability> implies <continuity>, whereas the converse need not hold.
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