| one sided derivative |
$f_{+}^{\prime}(0) = 1$ and $f_{-}^{\prime}(0) = -1$
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At $0$, the one-sided derivatives of $\abs{x}$ are $1$ and $-1$, and the corresponding Dini derivatives equal them.
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| clarke generalized gradient |
$\partial f(0) = [-1, 1]$
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The Clarke generalized gradient of $\abs{x}$ at $0$ is $[-1, 1]$, the convex hull of the nearby derivatives $\pm 1$.
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| subdifferential |
$\partial f(0) = [-1, 1]$
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The subdifferential of $\abs{x}$ at $0$ equals its Clarke generalized gradient, because $\abs{x}$ is convex.
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| weak derivative |
$f^{\prime}(x) = \operatorname{sign}(x)$
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The weak derivative of $\abs{x}$ is $\operatorname{sign}(x)$, because $\operatorname{sign}(x)$ satisfies the integration-by-parts formula for $\abs{x}$. The value of $\operatorname{sign}(x)$ at the single point $0$ does not matter.
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| idempotent |
✓ |
$x \mapsto \abs{x}$ is idempotent, because $\abs{\abs{x}} = \abs{x}$.
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| convex function |
✓ |
The function $x \mapsto \abs{x}$ is convex, because the segment between two points on its V-shaped graph lies on or above the graph.
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| differentiable |
✗ |
$x \mapsto \abs{x}$ is not differentiable at $0$, because it has a corner there.
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| continuously differentiable |
✗ |
$f(x) = \abs{x}$ is not continuously differentiable, because $f$ has no derivative at $0$, so the derivative is not a continuous function on all of $\reals$.
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| smooth |
✗ |
$f(x) = \abs{x}$ is not smooth, because it is not even differentiable at $0$.
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| analytic |
✗ |
$f(x) = \abs{x}$ is not analytic, because it is not even differentiable at $0$.
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| constant function |
✗ |
$x \mapsto \abs{x}$ is not constant.
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