# Linear interpolation of Lipschitz functions

Does the linear interpolation preserve the Lipschitz constant of a function? I mean, if $F:R^n \to R$ is Lipschitz continuous in $R^n$ with constant L, its linear interpolation obtained from some of its values has still the same Lipschitz constant? If yes, why?

• Have you considered the question for functions from $\Bbb R$ to $\Bbb R$, and drawn a picture? – John Hughes Aug 17 '17 at 20:29
• Yes, in that case it seems true to me. Is it always the case? I was trying to prove it analytically.... – Mimmo Aug 17 '17 at 21:00
• Can you prove it for that case? If so, that might help you prove it in the general case (if it's true). – John Hughes Aug 18 '17 at 10:51