$x \mapsto x \abs{x}$

Example · High school

Terms

Term	This example	Why
differentiable	✓	$x \mapsto x \abs{x}$ is differentiable at every $x$: away from $0$, the function equals $x^2$ or $-x^2$, and at $0$, the difference quotient $\frac{h \abs{h}}{h} = \abs{h}$ tends to $0$, so the derivative at $0$ is $0$.
continuously differentiable	✓	$x \mapsto x \abs{x}$ is continuously differentiable, because its derivative $x \mapsto 2 \abs{x}$ is continuous.
smooth	✗	$x \mapsto x \abs{x}$ is not smooth, because its derivative $x \mapsto 2 \abs{x}$ is not differentiable at $0$.
analytic	✗	$x \mapsto x \abs{x}$ is not analytic, because every analytic function is smooth.

Distinctions using this example

• Differentiable vs. continuously differentiable: $x \mapsto x \abs{x}$ shows that a continuously differentiable function can still have a derivative with a corner: the derivative $2 \abs{x}$ is continuous but not differentiable.

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