$|f(x,y)| \le K |x-y|$: is there a name for this property?

Question

Here $f:\mathbb{R}^2 \to \mathbb{R}$ and $K>0$ is a constant. It's a bit like being contractive or Lipschitz, but not the same as either of those.

Answer 1

One may compare this to the notion of a function with a modulus of continuity that is constant.

As @lisyarus points out, for a function $f:\mathbb R^2 \to \mathbb R$ to have a constant modulus of continuity $K \gt 0$ would mean:

$$ |f(x,y) - f(u,v)| \le K ||(x,y) - (u,v) || $$

whereas here we are told there exists some constant $K \gt 0$:

$$ |f(x,y)| \le K|x-y| $$

In particular this latter implies that the function $f(x,x)$ "restricted to the diagonal" is identically zero. We cannot infer this property simply from uniform continuity or other smoothness assumptions on $f(x,y)$.