[Math] Example for unbounded Lipschitz function on a bounded domain

Question

Let $D\subseteq \mathbb{R}^2$ be a bounded open rectangle in the plane.
(You can assume $D=(0,1) \times (0,1)$). Let $f:D \to \mathbb{R}$ be a continuous function which is uniformly-Lipschitz in the second variable $y$, i.e there exists $K>0$ such that

$$|f(x,y_2)-f(x,y_1)|\le K|y_1-y_2| \, \, \forall x,y_1,y_2 \in (0,1)$$

Is it true that $f$ is bounded on $D$?

Answer 1

No, simply choose an unbounded function that is continuous in the first coordinate and constant in the second, e.g. $\displaystyle f(x, y) = \frac{1}{x}$.