Subderivative

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 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.