These days, I'm working through Bredon's Topology and Geometry in order to get a better grasp on topology and fill in the gap of never having seen homology theory.
In chapter 14, Bredon proves:
If $f_0 \simeq f_1: X \to Y$ then $M_{f_0} \simeq M_{f_1} \operatorname{rel} X+Y$.
where $M_f$ is the mapping cone of $f$ (i.e. $X \times I \cup Y$ identifying $(x,0)$ with $f(x)$).
If I paraphrase, the mapping $M_{f_0} \to M_{f_1}$ is generated by "mapping the lower half of $X \times I$ to the homotopy (of $f_0 \simeq f_1$)" and similarly for $M_{f_1} \to M_{f_0}$.
The composition of these mappings should then be checked to be homotopic to the identity.
This seems easy enough, because under this mapping $X \times I$ partly maps onto a strip in $Y$, doubles up on itself, to end as the identity on the adjoining line $X \times \{0\}$. Thus one would say that we can "push" what is mapped to $Y$ back into the $X \times I$ part with ease.
Indeed, this agrees with what Bredon describes. However, he subsequently insists that the continuity of this homotopy be checked.
To me, this seems like a standard definition for a homotopy, and continuity seems glaringly obvious. Particularly in face of the standard lemma that a quotient map $X \times I \sqcup Y \to M_f$ yields a quotient map $(X \times I \sqcup Y)\times I \to M_f \times I$ (so that continuity may be checked on $X \times I^2$ and $Y\times I$).
Hence my question:
Is Bredon being formal by checking continuity, or should I be more careful in constructing homotopies because strange stuff may happen?
If the latter, an enlightening example would be highly valued.