One of the beneficial properties of persistent homology is its stability results (so called robustness to noise).

Usually the referenced paper is this paper titled "Lipschitz functions have $L_p$-stable persistence".

According to the authors, the Lipschitz condition is crucial (otherwise the results don't hold).

My understanding is that the Lipschitz function $f$ is used to construct the filtration $X_a = f^{−1}(−\infty, a] $ in the context of persistent homology.

My question is how "universal" is this assumption of Lipschitz-ness?

That is, for a "typical" filtration $K_1\subseteq K_2 \subseteq \dots K_m$, can it be expressed in the form of $K_i=f^{−1}(−\infty, a] $ for some Lipschitz function $f$?

If no, doesn't the oft-cited statement that persistent homology is "robust to noise" fail to hold?

Thanks a lot. That is my main question after reading the referenced paper since it is written in a very general viewpoint (more general than the context of persistent homology).