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Lipschitz Continuity

​
function
sin(x)
sin(x)
2
x
a
-1
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.
​
​
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