Example · High school
Terms
| Term | This example | Why |
|---|
| one sided derivative |
$f_{+}^{\prime}(0) = 1$ and $f_{-}^{\prime}(0) = -1$
|
|
| clarke generalized gradient |
$\partial f(0) = [-1, 1]$
|
The Clarke generalized gradient of $\lvert x \rvert$ at $0$ is $[-1, 1]$, the convex hull of the nearby derivatives $\pm 1$.
|
| subdifferential |
$\partial f(0) = [-1, 1]$
|
The subdifferential of $\lvert x \rvert$ at $0$ equals its Clarke generalized gradient, because $\lvert x \rvert$ is convex.
|
| weak derivative |
$f^{\prime}(x) = \operatorname{sign}(x)$
|
The weak derivative of $\lvert x \rvert$ is $\operatorname{sign}(x)$, because $\operatorname{sign}(x)$ satisfies the integration-by-parts formula for $\lvert x \rvert$. The value of $\operatorname{sign}(x)$ at the single point $0$ does not matter.
|
| idempotent |
✓ |
|
| differentiable |
✗ |
|
| continuously differentiable |
✗ |
|
| smooth |
✗ |
|
| analytic |
✗ |
|
| constant function |
✗ |
|
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