The derivative map is continuous

Fix a compact test set $C\subseteq TM$ in a weak neighbourhood of $df$. Cover $C$ by finitely many smaller compact pieces inside induced source bundle charts and target bundle charts containing their $df$-images; such pieces exist by small coordinate balls and compactness (Coordinate balls form a basis of a topological manifold, A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it). Their projections to $M$ are compact and their fibre coordinates $v$ are bounded.

In the coordinates of [L1], every derivative through order $r$ of $(f(x),J_f(x)v)$ is a derivative of $f$ through order $r+1$, multiplied at most by a bounded fibre coordinate, or an entry of a lower derivative after differentiating in $v$. Therefore sufficiently small weak errors in $f$ through order $r+1$ on the projected compact pieces imply all the order-$r$ conditions on $df$. Arbitrary total-space charts are handled by the atlas comparison The weak smooth topology is independent of the chosen atlas, whose chain-rule estimates apply on these compact pieces.

Intersect these finitely many neighbourhoods with the prescribed first-component neighbourhood of $f$. Its image under $f^{\mathrm{\prime }}\mapsto (f^{\mathrm{\prime }},d{f}^{\mathrm{\prime }})$ lies in the given product neighbourhood. Restricting to immersions proves continuity of $D$, without compactness of $M$.