Lipschitz continuity

Given two metric spaces (X, d_X) and (Y, d_Y), where d_X denotes the metric on the set X and d_Y is the metric on set Y (for example, Y might be the set of real numbers R with the metric d_Y(x, y) = |x − y|, and X might be a subset of R), a function

is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x₁ and x₂ in X,

^[1]

Any such K is referred to as a Lipschitz constant for the function ƒ. 
Reference：http://en.wikipedia.org/wiki/Lipschitz_continuity
区分概念，Lipschitz Continuously Differentiable，见Nesterov’s Optimal Gradient Method(Yaoliang Yu的ppt第十页)

[zz] 关于Lipschitz连续性的几个例子 http://blog.sina.com.cn/s/blog_544a70700100fqix.html
1，f(x) = |x|

是Lipschitz连续的，Lipschitz常数为1，但是其不可微

2，f(x) = x²

是Lipschitz不连续的，在无穷大处，f'(x)=2x,趋于无穷陡

3，f(x) = √x defined on [0, 1] is not Lipschitz continuous. This function becomes infinitely steep as x approaches 0 since its derivative becomes infinite.

4,The function f(x) = x^3/2sin(1/x) (x ≠ 0) and f(0) = 0, restricted on [0, 1], gives an example of a function that is differentiable on a compact set while not locally Lipschitz because its derivative function is not bounded. 

f'(x)=3/2√x sin(1/x)-cos(1/x)/√x 

趋向于0的时候无界