Example · High school
Terms
| Term | This example | Why |
|---|
| locally Lipschitz continuous |
✓ |
$x \mapsto \abs{x}$ on $[-1, 1]$ is locally Lipschitz continuous, because the function is Lipschitz continuous on all of $[-1, 1]$ with constant $1$, even at the corner at $0$.
|
| Lipschitz continuous |
✓ |
$x \mapsto \abs{x}$ is Lipschitz continuous with constant $1$, because $\abs{\abs{x} - \abs{y}} \leq \abs{x - y}$.
|
| Holder continuous |
✓ |
|
| uniformly continuous |
✓ |
$x \mapsto \abs{x}$ on $[-1, 1]$ is Hölder continuous (with $\alpha = 1$) and uniformly continuous, because it is Lipschitz continuous.
|
| absolutely continuous |
✓ |
$x \mapsto \abs{x}$ on $[-1, 1]$ is absolutely continuous, because it is Lipschitz continuous.
|
| continuously differentiable |
✗ |
$x \mapsto \abs{x}$ is not continuously differentiable, because it is not differentiable at $0$.
|
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