Let $f$ be the function defined as follows:
$$f:(t,x)\in R \times R \rightarrow f(t,x)= x^{\frac{2}{3}} \in R$$
I want to know whether $f$ is locally Lipschitz with respect to the variable $x$. Problem comes when studying the points $(t,0): t\in R$, because we cannot use the continuity of partial derivative with respect $x$ as it is not defined in those points. Thus, for a given neighbourhood of $(t,0)$, denoted by $U$, and $(t,x),(t,y) \in U$, I would like to prove whether the quotient:
$$\frac{|f(t,x)-f(t,y)|}{|x-y|} = \frac{|x^{\frac{2}{3}}-y^{\frac{2}{3}}|}{|x-y|}$$
is bounded or not. Any ideas?