Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 19 / a

The function $f$ is continuous at $x_0$ ifThe choice of $\delta$ may depend on $\epsilon$ and $x_0$, but must work for every $x$ in that neighbourhood. This is the epsilon-delta definition of continuity.

The function $f$ is differentiable at $x_0$ if there exists a finite real number $L$ such thatThen $f^{\prime }(x_0)=L$. The puncture $0<|x-x_0|$ makes the difference quotient defined; requiring a finite real $L$ distinguishes an ordinary derivative from an infinite slope.