Let $X,Y\subseteq \mathbb{R}$, I have read that if a function $f:X\to Y $ is CONTINUOUS and STRICTLY MONOTONE then is bijective from $X$ to $f(X)$.
My doubt is: the hypothesis of continuity is fundamental? I mean if a function is simply strictly monotone in $X$ then can I say that the function is bijective from $X$ to $f(X)$ and so $\exists f^{-1}:f(X)\to X$?