Lipschitz Continuity - Examples

Examples

Lipschitz continuous functions

• The function f(x) = √x² + 5 defined for all real numbers is Lipschitz continuous with the Lipschitz constant K = 1, because it is everywhere differentiable and the absolute value of the derivative is bounded above by 1.

• Likewise, the sine function is Lipschitz continuous because its derivative, the cosine function, is bounded above by 1 in absolute value.

• The function f(x) = |x| defined on the reals is Lipschitz continuous with the Lipschitz constant equal to 1, by the reverse triangle inequality. This is an example of a Lipschitz continuous function that is not differentiable. More generally, a norm on a vector space is Lipschitz continuous with respect to the associated metric, with the Lipschitz constant equal to 1.

Continuous functions that are not (globally) Lipschitz continuous

• The function f(x) = √x defined on is not Lipschitz continuous. This function becomes infinitely steep as x approaches 0 since its derivative becomes infinite. However, it is uniformly continuous as well as Hölder continuous of class C0, α for α ≤ 1/2.

Differentiable functions that are not (globally) Lipschitz continuous

• The function f(x) = x3/2sin(1/x) (x ≠ 0) and f(0) = 0, restricted on, gives an example of a function that is differentiable on a compact set while not locally Lipschitz because its derivative function is not bounded. See also the first property below.

Analytic functions that are not (globally) Lipschitz continuous

• The exponential function becomes arbitrarily steep as x approaches infinity, and therefore is not globally Lipschitz continuous, despite being an analytic function.
• The function f(x) = x2 with domain all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x approaches infinity. It is however locally Lipschitz continuous.