Let be X,Y
Banach spaces, f:X -> Y.
f is (locally)
Lipschitz continuous at x in X
iff it satisfies the
Lipschitz condition at x.
(There exists M > 0, M of
R and an
neighborhood U(x) of x with for all y of U(x) holds ||f(y)-f(x)|| < M ||x - y|| )
f is called Lipschitz continuous if it is Lipschitz continuous for all x of X.
Lipschitz continuous implies continuous.
Continuous differentiable implies Lipschitz continuous.