My topology textbook states that
the function $\mathrm h: X \rightarrow Y$ from topological space $\mathrm X$ to topological space $\mathrm Y$ is continuous if and only if the function $\mathrm k = i \times h: X \rightarrow X \times Y$ defined by $\mathrm k(x) = (x,h(x))$ for every $\mathrm x \in X$, is an homeomorphism of $\mathrm X$ with the subspace $\Gamma_h$ of $\mathrm X \times Y$
where $\mathrm i$ is the canonical inclusion and $\mathrm \Gamma_h = \{ (x,y) \in X \times Y | y = h(x) \}$.
I'm interested only in the necessary condition. The proof given is as follows:
Let i': $\mathrm \Gamma_h \rightarrow X \times Y$ be the canonical inclusion and k': $\mathrm X \rightarrow \Gamma_h$ the function such that k $\mathrm = i' \circ k': X \rightarrow X \times Y$. If $\mathrm k'$ is an homeomorphism, $\mathrm k$ is continuous, therefore $\mathrm h = q \circ k$ is continuous
where q: $\Gamma_h \rightarrow$ Y is the canonical projection.
I do understand every passage of the proof, but why do I have to introduce the homeomorphism $\mathrm k$'? If, by hypothesis, $\mathrm k$ is an homeomorphism, can't I just conclude that $\mathrm h$ is continuous because $\mathrm h = q \circ k$?