Editing Lipschitz continuity (section)

==One-sided Lipschitz==
Let ''F''(''x'') be an [[Semi-continuity|upper semi-continuous]] function of ''x'', and that ''F''(''x'') is a closed, convex set for all ''x''. Then ''F'' is one-sided Lipschitz<ref>{{cite journal |last1=Donchev |first1=Tzanko |last2=Farkhi |first2=Elza |year=1998 |title=Stability and Euler Approximation of One-sided Lipschitz Differential Inclusions |journal=SIAM Journal on Control and Optimization |volume=36 |issue=2 |pages=780–796 |doi=10.1137/S0363012995293694 }}</ref> if
:<math>(x_1-x_2)^T(F(x_1)-F(x_2))\leq C\Vert x_1-x_2\Vert^2</math> 
for some ''C'' and for all ''x''<sub>1</sub> and ''x''<sub>2</sub>.

It is possible that the function ''F'' could have a very large Lipschitz constant but a moderately sized, or even negative, one-sided Lipschitz constant. For example, the function

:<math>\begin{cases}
F:\mathbf{R}^2\to\mathbf{R},\\ 
F(x,y)=-50(y-\cos(x))
\end{cases}</math>

has Lipschitz constant ''K'' = 50 and a one-sided Lipschitz constant ''C'' = 0. An example which is one-sided Lipschitz but not Lipschitz continuous is ''F''(''x'') = ''e''<sup>−''x''</sup>, with ''C'' = 0.