DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02

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The left and right derivatives of a real function as one-sided limits of its difference quotient

Definition

Let $f:I\to\mathbb R$ and let $c\in I$ have points of $I$ on the indicated side. The left derivative and right derivative, when the corresponding one-sided limits exist as real numbers, are

$f'_-(c):=\lim_{x\to c^-}\frac{f(x)-f(c)}{x-c},\qquad f'_+(c):=\lim_{x\to c^+}\frac{f(x)-f(c)}{x-c}.$

These are the one-sided limits of the difference quotient (The left and right limits of $f$ at $c$ , as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$ ). If both exist and are equal, their common value is the ordinary derivative from The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set.

Depends on

Used by

A convex function on an open interval has finite left and right derivatives everywhere, with f'_-(u)≤ f'_+(u)≤ (f(v)-f(u))/(v-u)≤ f'_-(v)≤ f'_+(v) for u<v Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 24 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

R. Gardner, Convex Functions, Notes 6.6 (standard reference, not scraped)