# Lipschitz Continuity

Lipschitz Continuity

A function is Lipschitz continuous on an interval if there is a positive constant such that for all , in the interval. Geometrically this requires the entire graph of to be between the lines for any in the interval. The smallest possible is the largest magnitude of the slope of in the interval.

f(x)

m

|f(x)-f(x)|≤m|x-x|

2

1

2

1

x

1

x

2

f(x)

f(a)±m(x-a)

a

m

f(x)