Lipschitz continuity

For a Lipschitz continuous function, there exists a double cone (white) whose origin can be moved along the graph so that the whole graph always stays outside the double cone

We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line

Lipschitz continuous functions that are everywhere differentiable but not continuously differentiable

It is possible that the function F could have a very large Lipschitz constant but a moderately sized, or even negative, one-sided Lipschitz constant. For example, the function