Name of function satisfying property? Kind of Lipschitz

Question

Do functions satisfying the following property have a name?

$$\vert f(x_1,y_1) - f(x_2,y_2) \vert \leq K(x) \vert y_1-y_2\vert$$

where $x=\epsilon x_1 + (1-\epsilon)x_2$, for $\epsilon\in(0,1)$ fixed, and $\int K(x)^2dx<\infty$?

Answer 1

Your conditions imply $|f(x_1, y) - f(x_2, y)| \le 0$, hence, $f(x,y)$ is independent of $x$.

Then, you can take any $x$ with $K(x) < +\infty$ and get $|f(x, y_1) - f(x, y_2)| \le K(x) \, |y_1 - y_2|$.

Hence, $f$ is constant w.r.t. $x$ and Lipschitz w.r.t. $y$.