Lipschitz Continuity

function

sin(x)

sin(x)

x

2

a

-1

-3

-2

-1

1

2

3

-1.0

-0.5

0.5

1.0

A function

f(x)

is Lipschitz continuous on an interval if there is a positive constant

m

such that

|f(x

2

)-f(x

1

)|≤m|x

2

-x

1

|

for all

x

1

,

x

2

in the interval. Geometrically this requires the entire graph of

f(x)

to be between the lines

f(a)±m(x-a)

for any

a

in the interval. The smallest possible

m

is the largest magnitude of the slope of

f(x)

in the interval.