Cusp

At a cusp the one-sided derivatives become infinite, with opposite signs: the graph has two vertical half-tangents that “point” in opposite directions.

Definition — Cusp

$x_0$ is a cusp of $f$ if both one-sided derivatives tend to $\pm\infty$ with opposite signs: typically $f'_-(x_0) = -\infty$ and $f'_+(x_0) = +\infty$ (cusp pointing “downwards”) or vice versa.

Example — $f(x) = \sqrt{|x|}$ at $x_0 = 0$

For $x > 0$: $f(x) = \sqrt{x}$, $f'(x) = \frac{1}{2\sqrt{x}}\to +\infty$ as $x\to 0^+$. For $x < 0$: $f(x) = \sqrt{-x}$, $f'(x) = -\frac{1}{2\sqrt{-x}}\to -\infty$ as $x\to 0^-$. The one-sided derivatives are infinite and of opposite sign: a cusp pointing downwards at $(0,0)$.

At $x_0=0$ the two half-tangents are vertical and point in opposite directions: the cusp of $\sqrt{|x|}$.

Links

Topics: Derivatives
Concepts: Cusp · One-sided derivative · Non-differentiable point
Methods: Non-differentiable point cusp
Skills: Calculating limits