Is the function $f(x)=x^2$ absolutely continuous on the real line?

Question

In Wiki (Lipschitz), it says:

A Lipschitz function $g : \mathbb{R}\to \mathbb{R}$ is absolutely continuous.

According to the definition of absolute continuity, I am confused about an simple example:

$$f(x) = x^2$$

This is not a Lipschitz function since we cannot find a bounded $L$ such that

$$|f(y) - f(x)| \leq L||y-x||_2$$

But is $f(x)$ absolutely continuous? (I am confused about if $x$ approaches $\infty$)

Answer 1

For function on the real line, one has the following chain of inclusions (adapted from Wikipedia, which considers only a compact subset):

Differentiable with a bounded derivative $\subset$ Lipschitz continuous $\subset$ absolutely continuous $\subset$ uniformly continuous $\subset$ continuous

None of the inclusions are difficult to prove: use, left to right,

1. Mean Value Theorem
2. Triangle inequality
3. Definition of absolute continuity specialized to one interval
4. Definition of uniform continuity

All inclusions are strict, by the way.

The function $f(x)=x^2$ is not uniformly continuous on $\mathbb{R}$, and therefore is not absolutely continuous.