Prove that $y \mapsto f(x,y)$ is Lipschitz continuous, where $$f(x,y) = \frac{y}{x} \ln{\frac{y}{x}}, \ \ \ |x-1| \leq \frac{1}{2}, |y-1| \leq \frac{1}{2}e$$
I tried to solve this, but I find it very difficult to do this in two variables. How do I start solving this?
My first attempt was trying to solve this for the four cases ($x > \frac{3}{2}, y > \frac{1}{2}e + 1$) etc. But this didn't work out very well.