$\text{sign}(x)$ and one-sided Lipschitz

Question

We say that a function $f:R^2\longrightarrow R$ satisfies one-sided Lipschitz condition with respect to x with constant $K$ if $$\langle f(x_{1},y)-f(x_{2},y),x_{1}-x_{2}\rangle \leq K||x_{1}-x_{2}||^2.$$ And my question is how can we check if this these functions $f(x,y)=-\text{sign}(x)$ $f(x,y)=\text{sign}(x)$ satisfy one-sided Lipschitz condition or not.

Answer 1

Hint. A useful approximation is this: $$\text{sign} (x)=\lim_{t \to 0} \frac{x}{\sqrt{x^2+t^2}}$$

Do you see how to continue?

Or you can just consider the two cases.

First: $$\text{sign} (x_1)=\text{sign} (x_2) \\ | \text{sign} (x_1)-\text{sign} (x_2)|=0$$

Second: $$\text{sign} (x_1)=-\text{sign} (x_2) \\ | \text{sign} (x_1)-\text{sign} (x_2)|=2$$

Do you see how to continue here?