We prove the max-martingale conjecture given in recent article with Marc Yor. We show that for a continuous local martingale $(N_t:t\ge 0)$ and a function $H:R x R_+\to R$, $H(N_t,\sup_{s\leq t}N_s)$ is a local martingale if and only if there exists a locally integrable function $f$ such that $H(x,y)=\int_0^y f(s)ds-f(y)(x-y)+H(0,0)$. This implies readily, via Levy's equivalence theorem, an analogous result with the maximum process replaced by the local time at 0.