Definition of a piecewise function

A piecewise function (or piecewise-defined function) is a function that changes “formula” depending on which interval $x$ falls into. We have already met them in the study of the absolute value: now we treat them as an object in their own right.

Definition — Piecewise function

A function $f\colon D\to\mathbb{R}$ is piecewise defined if $D$ is split into $n$ intervals $I_1,\ldots,I_n$ with $D=I_1\cup\cdots\cup I_n$ and on each $I_k$ a formula $f_k(x)$ is assigned: $f(x) = \begin{cases} f_1(x) & x\in I_1, \\ f_2(x) & x\in I_2, \\ \vdots & \\ f_n(x) & x\in I_n. \end{cases}$

Example — Sign function

$\operatorname{sgn}(x) = \begin{cases} +1 & x>0, \\ 0 & x=0, \\ -1 & x<0. \end{cases}$ Three constant pieces. It is not continuous at $x=0$: a jump of height $2$.

Example — Absolute value as a piecewise function

$|x|=\begin{cases} x & x\ge 0 \\ -x & x<0\end{cases}$. Two linear pieces meeting at $(0,0)$ at the same height: continuous but not differentiable at $0$ (a corner point, which we shall see in Year Five).

Links

Topics: Functions and properties
Concepts: Piecewise function · Sign function · Absolute value
Functions: Sign function · Absolute value