Lipschitz continuity

In mathematics, a function

Examples

• The function $f(x)=x^2$ defined on $$ is Lipschitz continuous, with K=14. This follows from the observation above.
• The function $f(x)=sqrt\{x^2+5\}$ defined for all real numbers is Lipschitz continuous with the Lipschitz constant K=1.
• The function $f(x)=2|x-3|$ defined on $$ is Lipschitz continuous with the Lipschitz constant equal to 2. This is an example of a Lipschitz continuous function which is not differentiable.
• The function $f(x)=x^2$ (the same function as in the first example) defined for all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x→∞.
• The function $f(x)=sqrt\{x\}$ defined on $$ is not Lipschitz continuous. This function becomes infinitely steep as x→0 since its derivative becomes infinite.