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The left and right derivatives of a real function as one-sided limits of its difference quotient
Definition
Let and let have points of on the indicated side. The left derivative and right derivative, when the corresponding one-sided limits exist as real numbers, are
These are the one-sided limits of the difference quotient (The left and right limits of at , as limits of the restrictions of to and ). If both exist and are equal, their common value is the ordinary derivative from The derivative of at a point that is a limit point of , and differentiability on a set.
Depends on
- The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$
- 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
Used by
Dependency tree · two levels
11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- R. Gardner, Convex Functions, Notes 6.6 (standard reference, not scraped)