---
tag: A289X08MR1
kind: definition
name: Subderivative
msc: [26A51, 49J52, 52A41]
defines: 9C5WXYYQEK
uses: [0TK8Y3J5RW, 0VV0HSW1ZS, 0F1WR9C83K]
sources:
  - url: https://en.wikipedia.org/wiki/Subderivative
---

## Definition

Let $I$ be an open [[interval|0TK8Y3J5RW]] of $\R$, let $f \colon I \to \R$ be a [[convex function|0VV0HSW1ZS]], and let $x_0 \in I$.
A real number $c$ is a *subderivative* of $f$ at $x_0$ if
$$f(x) - f(x_0) \ge c \, (x - x_0) \quad \text{for all } x \in I.$$
The *subdifferential* $\partial f(x_0)$ is the set of all subderivatives of $f$ at $x_0$.

## Remarks

The subdifferential is the closed interval $\partial f(x_0) = [f'_-(x_0), f'_+(x_0)]$, where $f'_-$ and $f'_+$ are the one-sided derivatives, which exist for a convex function on an open interval.
In particular it is never empty.
It is the single point $\set{f'(x_0)}$ exactly when $f$ has a [[derivative|0F1WR9C83K]] at $x_0$.
For a convex function on $\R^n$, a vector $g$ with $f(x) - f(x_0) \ge \ip{g}{x - x_0}$ for all $x$ is called a *subgradient*.

Example: $f(x) = \abs{x}$ on $\R$ and $x_0 = 0$.
For $c \in [-1, 1]$ one has $\abs{x} \ge c x$ for all $x$, so $\partial f(0) = [-1, 1]$.

Non-example: $c = 2$ for the same $f$ and $x_0$.
At $x = 1$ the condition reads $1 \ge 2$, which fails.