A Lipschitz continuous function is a function that must be continuous and almost differentiable everywhere. So, for any function to be a Lipschitz continuous, the absolute value of a slope of the function’s graph cannot be more than a constant . This constant is called the Lipschitz constant:

To put it in simple terms, we can say a function is Lipschitz continuous when the derivation of a function is bounded by some constant K and it never exceeds the constant.

Let's say , for instance,  is Lipschitz continuous since its derivative is bounded by 1. Similarly,  is Lipschitz continuous, since its slope is -1 or 1 everywhere. However, it is not differentiable at 0.

So, let's recall our equation:

Here, supremum is basically an opposite to infimum. So, supremum over the Lipschitz function implies a maximum over k-Lipschitz functions. So, we can write the following:

The preceding equation basically tells us that we are basically finding a maximum distance between the expected value over real samples and the expected value over generated samples.