= Solution
\b[<Continuity>.] The <function> is <continuous> at $x_0$ when $\lim_{x\to x_0}f(x)=f(x_0)$. Equivalently, for every $\varepsilon>0$ there exists $\delta>0$ such that $|x-x_0|<\delta$ within the <domain of a function> implies $|f(x)-f(x_0)|<\varepsilon$; this is the <epsilon-delta definition of continuity>.
\b[<Differentiability>.] At an interior point $x_0$, the <difference quotient> must have a finite two-sided <limit of a function>:
$$
f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}h.
$$
This defines <differentiability> and the <derivative>. At an endpoint one may instead state a one-sided definition relative to the <interval>. <Differentiability implies continuity>, because the increment is $h$ times a quotient with finite limit, but <continuity> alone does not guarantee <differentiability>.
Back to article page
