How to show that the set of Lipschitz functions is uniformly bounded?

Question

I want to show that the set of Lipschitz functions $f$

$$|fp-fq|\leq Ld(p,q)$$

on a compact domain is uniformly bounded.

I don't see a way to do this without any additional condition. Is this even possible?

Answer 1

This statement is clearly false, because adding a constant to any Lipschitz function preserves the Lipschitz condition, and this can be done with arbitrary large constants.

In other words, let $g$ be any $L$-Lipschitz function on $X$. Then $g_{n}=g+n$ satisfies \begin{align*} |g_{n}(x)-g_{n}(y)|=|g(x)+n-g(y)-n|=|g(x)-g(y)|\leq Ld(x,y) \end{align*} for all $x,y\in X$, so $g_{n}$ is $L$-Lipschitz for every $n$. And the sequence $(g_{n})_{n=1}^{\infty}$ is not uniformly bounded because $g_{n}(x)=g(x)+n\to\infty$ as $n\to\infty$ for any $x\in X$.