# Strategies to prove Lipschitz

Can someone share what usual strategies they pursue for proving Lipschitz? I usually use two strategies but they are very restrictive on the type of functions they require. One is by checking the bound of the continuous derivative, the other is by use of the mean value theorem. Unfortunately the mean value theorem cannot be used in Rn.

Also, the function I'm working is of the type: $\int_0^t f(x(s),y(s))ds$

I am looking for Lipschitz in both $(x(s),y(s))$ with the integrand being bounded by a function of $s$. Lipschitz here does not need to be a constant but may be a function of $t$.

• In general, just use the definition. (Not intended to be sarcastic.) As an analogy, showing Lipschitz by using the fact that the derivative is bounded is like using short cut rules for differentiating functions (power rule, product rule, etc.). The short cut rules work when you're in relatively nice situations, but for something like $f(x) = x^{2}\sin(1/x)$ with $f(0)=0,$ you have to use the definition of the derivative (as a limit). In general, there's going to be some inequality "dirty work" involved in showing something is (or is not) Lipschitz when differentiability isn't available. – Dave L. Renfro May 15 '14 at 14:55