Absolute value is continuous but not differentiable at 0

Builds on One-sided derivative Differentiable functions are continuous

Counterexample

The function $f(x) = |x|$ is continuous on $\mathbb{R}$ , but it is not differentiable at $0$ . The difference quotient is $\frac{|h|}{h}$ , which is $1$ for $h > 0$ and $-1$ for $h < 0$ . So it has no limit as $h \to 0$ .

Remarks

The one-sided derivatives at $0$ exist: the right derivative is $1$ and the left derivative is $-1$ . So the converse of differentiable functions are continuous fails. For $x \ne 0$ , the derivative is $x / |x|$ , the sign of $x$ . Mathlib states the failure at $0$ as not_differentiableAt_abs_zero.

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Limit point of a set definition
Limit of a function definition
Continuity at a point definition
Derivative definition
One-sided limit definition
Differentiable function definition
One-sided derivative definition
Differentiable functions are continuous statement
Absolute value is continuous but not differentiable at 0 counterexample

Tag	`0CQ9ZM383N`
URI	`m2://0CQ9ZM383N`
Kind	counterexample
Writing	W0 stub
MSC	26A24
Uses	One-sided derivative, Differentiable functions are continuous

Absolute value is continuous but not differentiable at 0

Counterexample

Remarks

Reading path

Further reading

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