I am studying the article Immersion Theory for Homotopy theorists by Michael Weiss for my bachelor thesis. The main theorem states that the space of immersions and formal immersions between two smooth manifolds $M$ and $N$ are weakly homotopy equivalent, with the weak equivalence given by the map $$ \text{imm}(M,N)\to \text{fimm}(M,N),\quad f\mapsto (f,\mathrm{d}f). $$
In the article he never mentions continuity of this map, and maybe its really trivial, but I'm having some difficulty proving this. For the first component this is almost by definition of the $C^\infty$ topology, but for the second component I can't work it out.
Could anyone help me out?
Edit: The definition I have for the weak (compact-open) $C^r$ topology is the following: For a $C^r$ map $f$ and a compact set $K$ such that $f(K)\subseteq V$ define a $\textit{weak subbasic neighbourhood}$ around $f$ to be the set of $C^r$ maps $g: M \to N$ such that $g(K) \subseteq V$ and $$ ||D^k(\psi f \varphi^{-1})(x) - D^k(\psi g \varphi^{-1})(x)|| < \varepsilon, $$ which we denote by $N^r(f, (\varphi, U), (\psi,V),K,\varepsilon)$.
Second edit: Following the comments from @ronno, here's my attempt.
For conveniency we denote the map $\text{imm}(M,N)\to \text{fimm}(M,N)$ with $\delta$. Then we inspect the pre-image of some subbasic neighborhood $N^0(df, (U,\varphi), (V,\psi), K, \varepsilon)$. As I see it, we have: $$ \delta^{-1}(N^0(\dots))= \{g\in C^1(M,N)\mid \forall x\in \varphi(K): \quad ||D(\psi g \varphi^{-1})(x)-D(\psi f \varphi^{-1})(x)||<\varepsilon \} $$ I suppose from here I should that every point admits a subbasic neighborhood contained in this pre-image. I'm guessing it should be something of the form $N^1(g, (\pi_M(U), ?), (\pi_N(V),?), \pi_M(K), \varepsilon/3)$ but I don't know what the charts would have to be to make this work.