Let $f(x,y)=y^{\frac{2}{3}}$. I want to show that functions is Lipschitz function w.r.t $y$ on $\mathbb{R}^2$ or not.
Given definition of function that satisfy Lipschitz conditions w.r.t to $y$, i.e.
A function $f(x,y)$ defined on a (open or closed) domain $D$ is said to satisfy Lipschitz conditions with respect to $y$ for the constant $K>0$ if for every $x, y_1,y_2$ such that $(x,y_1),(x,y_2)$ are in $D$ $$|f(x,y_1)-f(x,y_2)|≤K|y_1-y_2|$$
I tried to do by using that definition. This is my attempt: $|f(x,y_1)-f(x,y_2)|=|y_1^{2/3}-y_2^{2/3}|≤K|y_1-y_2|$
Then I don't know how what must I do next.
Any help will be appreciated.