It is useful because it enables us to quantify our assumption that asmall change in the input made by an algorithm such as gradient descent will have a small change in the output.

University of Michigan - Ann Arbor

A function $$f$$ is Lipschitz continuous if there exists a non-negative real constant $$L$$ (referred to as the Lipschitz constant) such that for all $$\mathbf{x}$$ and $$\mathbf{y}$$ in its domain:

$$|f(\mathbf{x}) - f(\mathbf{y})| \le L \|\mathbf{x} - \mathbf{y}\|_2$$

This condition bounds how rapidly the function's value can change relative to the change in its inputs.

Lipschitz Continuous Function

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

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