Can anyone prove that the product of two Lipschitz functions is itself Lipschitz? ( Lipschitz : a function f such that for all x,y in D, there exists an L > 0 s.t. d( f(x), f(y) ) <= L*d( x,y ) )
This is not true: take $\displaystyle \vert \cdot \vert :\mathbb{R} \rightarrow \mathbb{R}$ (absolute value) is Lipschitz but $\displaystyle x^2$ is not. If $\displaystyle D$ is bounded however use $\displaystyle \vert f(x)g(x) - f(y)g(y) \vert \leq \vert f(x) \vert \vert g(x)-g(y) \vert + \vert g(y) \vert \vert f(x)-f(y) \vert$