Is it true that a monotic, differentiable function with non-zero derivative has a continuous inverse? Is it true that a strictly monotic, differentiable function on $\mathbb{R}$ with non-zero derivative has a continuous inverse? This is a small caveat in a problem I'm working on, if this is true then I'm all good.

Something stronger is true. A strictly monotone and continuous function on $\mathbb R $ has a continuous inverse. The function doesn't even have to have a derivative.