Αν εχω μια ανω φραγμενη παραγωγισιμη συναρτηση τοτε αυτη ειναι Lipschitz (αυτο προκυπτει απο το Θ.Μ.Τ.), και στη συγκεκριμενη ειναι: $\frac {\partial f} {\partial y}=-2y$, οποτε αν εχω ορισει την f στο ορθογωνιο $0\leq x \leq1 , 0\leq y \leq1$ τοτε προκυπτει: $\frac {\partial f} {\partial y}=-2y \Rightarrow | \frac {\partial f} {\partial y}|=|-2y| \leq2$ , συνεπως η f ειναι Lipschitz με σταθερα Lipschitz k=2
Δες και εδω
http://en.wikipedia.org/wiki/Lipschitz_continuity : Lipschitz continuity of functions on the real line is closely related to differentiability. An everywhere differentiable function g : R → R is Lipschitz continuous (with K = sup |g′(x)|) if and only if it has bounded first derivative; one direction follows from the mean value theorem. In particular, any C1 function is locally Lipschitz, as continuous functions on a locally compact space are locally bounded.