Piecewise Functions — Cheat sheet

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Key method

To evaluate, first decide which condition the input satisfies, then use only that formula. At a boundary, inspect strict and non-strict inequality signs to determine endpoint inclusion.

Example

Let $f(x)=x+2$ for $x<0$ and $f(x)=x^2$ for $x\ge0$ . Then $f(-1)=1$ , $f(0)=0$ , and $f(2)=4$ . The left branch approaches $2$ near zero, while the value at zero is determined by the second branch. This is a jump, despite both formulas being individually continuous.

Avoid this mistake

Using both formulas at a boundary without checking the case conditions can assign two incompatible outputs.