Using the property "onto" in a proof Definition (Minimal Open Set): A nonempty open set $U$ of $X$ is said to be a minimal open set if and only if any open set which is contained in $U$ is $∅$ or $U$.
Definition (Minimal continuous map): Let $X$ and $Y$ be the topological spaces. A map $f: X → Y$ is called minimal continuous if $f^{-1}(M)$ is an open set in $X$ for every minimal open set $M$ in $Y$.
Definition (T$_{min}$ Space): A topological space $(X,\tau)$ is said to be T$_{min}$ space if every nonempty proper open subset of $X$ is minimal open set.
Theorem: Let : $f:X\rightarrow Y$  be a minimal continuous, onto map and Y be a T$_{min}$ space.
Then $f$ is continuous.
Proof: Let be $f$ a minimal continuous, onto map. Let $N$ be any nonempty proper open set in $Y$. By
hypothesis, $Y$ is T$_{min}$ space. It follows that $N$ is a minimal open set in $Y$. Since $f$ is minimal continuous, $f^{-1}(N)$ is an open in X. Therefore is a continuous.
Now, where did we use that $f$ is onto? I think if $f$ were not be onto, the proof would still work.
 A: In order to show that $f$ is continuous, we need to show that every open subset $O$ of $Y$ has an open inverse image.
But $f^{-1}[Y]=X$ for any function $f:X \to Y$, by definition and $X$ is always open in $X$. Likewise $f^{-1}[\emptyset] = \emptyset$ always and $\emptyset$ is always open in $X$, so the open sets $O=\emptyset,Y$ have been taken care of.
So we only need to consider $O$ with $\emptyset \neq O \neq Y$. But then indeed as $Y$ is $T_{\text{min}}$ by definition $O$ is minimal open in $Y$ and then as $f$ is "minimal continuous" we also have  that $f^{-1}[O]$ is open in $X$.
So always $f^{-1}[O]$ is open for all $O \subseteq Y$ open. So $f$ is continuous.
I don't see where ontoness would be needed, maybe the author thought it would be needed for $f^{-1}[Y]=X$ but this is not the case! $f^{-1}[Y] = \{x \in X\mid f(x)\in Y\} = X$ as $f$ maps $X$ into $Y$ by definition. Ontoness is saying $f[X]=Y$.
FWIW: I think these concepts of $T_{\text{min}}$ spaces and minimally continuous maps are a bit trivial and unnatural. There are hardly interesting natural examples. Nothing in "real life" fulfills it. The indiscrete space and some trivial variations thereof. Nothing worth bothering about. My 2 cts.
