Graphing Piecewise Functions — Cheat sheet

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Key method

At each boundary, record the left-hand behavior, the right-hand behavior, and the function value separately. They can be three different things. A function may be defined at a boundary without being continuous there.

Example

Let $f(x)=x+1$ for $x<1$ and $f(x)=3-x$ for $x\ge1$ . Draw the first line only left of $1$ and the second only on and right of $1$ . Both approach $2$ , and the second assigns $f(1)=2$ , so the graph joins continuously. The left slope is $1$ and right slope is $-1$ , so there is a corner.

Avoid this mistake

Continuity does not require the same formula or slope on both sides. Conversely, differentiability requires more than the pieces simply meeting.