DefinitionDefinition: AI-adaptedProof: Not applicableaudited 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 $f : I \to R$ 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

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

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 $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set.

Depends on

Dependency tree · 2 levels

Sources

R. Gardner, Convex Functions, Notes 6.6