Prove that $f(x,y)$ which satisfies: $$|f(x,y)-f(x,z)|\leq h(x)|y-z|$$ $$h(x): (0,a]\rightarrow \mathbb{R}, \, \text{integrable function}$$ is Lipschitz with respect to $y$.
Attempt:
For $f$ to satisfy the Lipschitz condition, it must be shown that $$h(x) \leq k$$ for some positive constant $k$.
I know that functions which are continuous on a closed bounded interval are bounded, but I'm far from proving the continuity of $h$, plus its domain is not a closed interval.