In kernel regression, what are the common theoretical motivations for using a kernel that is Lipschitz continuous?

Question

I read a few papers on Nadaraya-Watson kernel regression in which I saw assumptions that require the kernel function being Lipschitz continuous without explanation ( and without citation of such assumption in the proof), so I'm wondering what are some common reasons for imposing Lipschitz continuity condition on the kernel (instead of just being continuous). An illustration with an example would be great! Thanks!