While reading about various results related to density of smooth functions in the space of continuous functions with strong topology, I've got the impression that it is a general fact that for any continuous function $f:X\to Y$ between smooth manifolds, there is a strong neighbourhood $\mathcal U$ of $f$ which is contained in homotopy class of $f$, that is, for any $g\in\mathcal U$ there is a homotopy between $f$ and $g$.
It would also be a very nice result in any case, but I've yet to find the fact spelled out in clear terms, neither did I find a proof.
I would appreciate a hint or a reference to literature.
Notes:
- This question is closely related (pardon the pun) to the question Are close maps homotopic?, but this is different in that I allow arbitrary strong neighbourhoods (and the counterexamples listed there do not work in this more general context).
- It is also related to a question of mine: Are locally homotopic functions homotopic? – I believe that we can show that there is a strong neighbourhood of $f$ which is contained in „local homotopy class” in the sense explained in the question, using local convex structure of $Y$.
- A different idea of a proof would be showing that any continuous function has a contractible neighbourhood in the space of continuous functions by some abstract argument, but I'm not sure about the technical details of that.
Edit: I think I've managed to prove the fact, but the proof is somewhat long and I'm rather tired right now, so I won't write it down right now. If someone is really curious I probably might give a sketch. In any case, a reference would probably be the best answer, unless someone knows a simple proof of this. :)