Continuous or Differentiable but Nowhere Lipschitz Continuous Function

Question

1. What is a real valued function that is continuous on a close interval but not Lipschitz continuous on any subinterval?

2. What is a real valued function that is differentiable on a close interval but not Lipschitz continuous on any subinterval?

Answer 1

Continuous and nowhere Lipschitz

An example is given by the Weierstrass function, which is continuous and nowhere differentiable. This can be justified in two ways:

• A Lipschitz function is differentiable almost everywhere, by Rademacher's theorem.

• Direct inspection of the proof that the function is nowhere differentiable; the estimates used in the proof also imply it's nowhere Lipschitz ($|f(x+h)-f(x)|$ is estimated from below by $|h|^\alpha$ with $\alpha<1$, for certain $x,h$).

Differentiable and nowhere Lipschitz

There are no such examples: a differentiable function on an interval must be Lipschitz on some subinterval. The following is an adaptation of a part of PhoemueX's answer.

1. The function $f'$ is a pointwise limit of continuous functions: namely, $$f'(x) = \lim_{n\to\infty} n(f(x+1/n)-f(x))$$ where for each $n$, the expression under the limit is continuous in $x$.

2. Item 1 implies that $f'$ is continuous at some point $x_0$. This is a consequence of a theorem about functions of Baire class 1: see this answer for references.

3. Continuity at $x_0$ implies $f'$ is bounded on some interval $(x_0-\delta,x_0+\delta)$. The Mean Value Theorem then implies that $f$ is Lipschitz on this interval.