TITLE: Homotopic maps which aren't relative homotopic
QUESTION [7 upvotes]: Can anyone give an example of two continuous maps, let say $f,g \colon X \to Y$, such that the set $A =\{ x \in X \mid f(x) = g(x) \}$ is not empty, the two maps are homotopic but there's no homotopy between these function relative to any non empty subspace of $A$?

REPLY [2 votes]: Actually I recently come back to this problem and I've found myself another counterexample. 
Consider the space 
$$X= \{ (x,y) \in \mathbb R^2| (x \ne 0 \land y/x \in \mathbb Q) \lor x=0\}$$
this is the space formed by the straight lines with rational slope.
This space can be deformation retracted on the point $(0,0)$, I'm going to call $p \colon X \to X$ the constant map sending $X$ in this point, and so the identity is relative homotopic to $p$.
The point $(0,0)$ is the only point to which the whole space can deformation retract.
Clearly every other constant map is homotopic to this map, so every other constant map is homotopic to the identity. 
Any constant map have a unique fixed point, which clearly is fixed by the identity too. The identity cannot be homotopy equivalent to any constant map 
relatively to the fixed point, with the exception of $p$, because otherwise the relative homotopy would be a deformation retraction of $X$ on this other point, which is not possible for said above.