Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 19 / a / i

For statement (1), continuity at the interior point meansExplicitly, for every $\epsilon >0$ there exists $\delta >0$ such that $|x-x_0|<\delta$ implies $|f(x)-f(x_0)|<\epsilon$ 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:The same finite value must be obtained from positive and negative $h$. Differentiability implies continuity, whereas the converse need not hold.