Geometry of generalized Lipschitz functions

Question

Clearly, that geometry of Lipschitz function is a double cone whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone (image from wiki).

I've found the definition of generalized Lipschitz functions:

Function f is said to be generalized Lipschitz if there exist L > 0 such that |f(x)-f(y)| ≤ L(1+|x-y|), ∀x,y

But a'm really stuck - what is the geometry of 2D generalized Lipschitz functions?

Answer 1

For $x$ fixed $$ -f(x)-L(1+|x-y|)\le f(y)\le f(x)+L(1+|x-y|),\quad \forall y. $$