Does an idempotent function has to be 1-Lipschitz?

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A function $f$ satisfying $f(f(x)) = f(x)$ for all $x$ is called an Idempotent function. I want to know if it has to be a Lipschitz function. Actually, I am training a neural network which has to be idempotent. Now I need to know whether imposing Lipschitz regularization can improve my results.