In Mathematical/High Dimensional Statistics:
One fairly amazing result is for $X=(X_1,\dots,X_n)$ where the coordinates are i.i.d. standard Gaussians, and $f:R^n \to R$ a $L$-Lipschitz function (w.r.t. Euclidean norm), then the random variable $f(X) - E[f(X)]$ is sub-Gaussian with sub-Gaussian norm $L$, i.e. it satisfies for any $\epsilon \ge 0$: $$ P(|f(X)- E[f(X)]| \ge \epsilon) \le 2 e^{-\epsilon^2/2 L^2}. $$ It is somewhat surprising (to me at least) how often this result comes in handy - particularly when working with rademacher/gaussian complexities in mathematical statistics/learning theory and also in random matrix analysis.
To give one concrete example, if $X$ is a $n \times p$ random matrix with i.i.d. standard gaussian entries, and $Y$ is another matrix, then by Weyl's inequality:
$$ \max_{k=1,\dots, p} |\sigma_k(X) - \sigma_k(Y)| \le \|X-Y \|_F, $$ where $\sigma_k$ is the $k$-th largest singular value. This tells us that the $k$-th singular value is a 1-Lipschitz function of the matrix, and so we immediately get $$ P(|\sigma_k(X) - E[\sigma_k(X)]| \ge \epsilon) \le 2e^{-\epsilon^2/2}. $$
I would suggest the following two books for more on this kind of usage of Lipschitz continuity:
- High Dimensional Statistics - A non-asymptotic viewpoint by Martin J Wainwright
- High Dimensional Probability by Roman Vershynin
In machine learning:
The concept of Lipschitz continuity is also important in the current machine/deep learning literature where model robustness (which they refer to as adversarial robustness) is an important issue. To understand the robustness of a complicated model (such as a neural network), a lot of work has gone into training networks that define an input-output map with small Lipschitz constant. The intuition here is that if my model is robust, it should not be too affected by perturbations in the input, $f(x+\delta x) \approx f(x)$, and this would be ensured by having $f$ be L-Lipschitz where $L$ is small. See this paper Training robust neural networks using Lipschitz bounds and their references for more on this.