Almost surely, we can write for every $t \ge 0$, $M_t=M_0+\beta_{\langle M,M \rangle_t}$, where $\beta$ is some Brownian motion. 

By Kahane's theorem, almost surely, for every $s \ge 0$, $\limsup_{\delta \to 0} \delta^{-1/2}|\beta_{t+\delta}-\beta_t| \ge 1$. 
https://www.ams.org/journals/tran/1986-296-02/S0002-9947-1986-0846605-2/S0002-9947-1986-0846605-2.pdf

Let $b > a \ge 0$. Almost surely, on the event $M$ is differentiable on the time interval $[a,b[$, we derive $d\langle M,M \rangle_t/dt = 0$ on the time interval $[a,b[$, hence $\langle M,M \rangle_t$ and $M_t$ do not depend on $t$ on the time interval $[a,b[$.