One-sided derivative

Definition

Let $f$ be a real function defined on an interval $[a, a + r)$ with $r > 0$ . The right derivative of $f$ at $a$ is the one-sided limit $f'_+(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h},$ when it exists. The left derivative $f'_-(a)$ is defined in the same way, for $f$ on $(a - r, a]$ and $h \to 0^-$ .

Remarks

A function on an open interval is differentiable at $a$ if and only if both one-sided derivatives exist and are equal. Their common value is the derivative. One-sided derivatives give a meaning to derivatives at the endpoints of a closed interval. For $|x|$ at $0$ , the right derivative is $1$ and the left derivative is $-1$ . Mathlib expresses them with HasDerivWithinAt on a half line.

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Limit point of a set definition
Limit of a function definition
Derivative definition
One-sided limit definition
One-sided derivative definition

Where this leads

5 pages build on this one.

Tag	`0HXEX0DPRA`
URI	`m2://0HXEX0DPRA`
Aliases	`left-derivative`, `right-derivative`
Kind	definition
Writing	W0 stub
MSC	26A24
Defines	One-sided derivative
Uses	Derivative, One-sided limit

One-sided derivative

Definition

Remarks

Reading path

Where this leads

Further reading

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