Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| Holder continuous |
✓ |
The Cantor function is Hölder continuous with $\alpha = \log 2 / \log 3$, and so it is uniformly continuous.
|
| uniformly continuous |
✓ |
The Cantor function is uniformly continuous because it is continuous on the compact interval $[0, 1]$.
|
| continuous |
✓ |
The Cantor function is continuous, and uniformly continuous because $[0, 1]$ is compact.
|
| Lipschitz continuous |
✗ |
The Cantor function is not Lipschitz continuous, because it rises by $2^{-n}$ over intervals of length $3^{-n}$, so its slopes are unbounded.
|
| absolutely continuous |
✗ |
The Cantor function is not absolutely continuous: it rises from $0$ to $1$, but its derivative is $0$ almost everywhere, so it is not the integral of its derivative.
|
| continuously differentiable |
✗ |
|
Distinctions using this example
All examples · All distinctions · All definitions