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I currently study the above mentioned theorem for jets. Just for convenience here is the statement:

Let $X,Y$ be smooth manifolds and $W \subset J^k(X,Y)$ a submanifold of the k-jet space. Define $T_W = \{f \in C^{\infty}(X,Y) ~|~ j^kf \pitchfork W\}$. Then $T_W$ is a residual subset of $C^{\infty}(X,Y)$ in the strong topology.

I found this theorem in Hirsch - Differential Topology and in Golubitsky/Guillemin - Stable mappings and their singularities and they both stated the theorem for smooth manifolds and maps. I just wonder how essential it is to work with smooth maps and manifolds. Does anyone now a version of this theorem for $C^r$ map and manifolds or is this simply not possible?

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After some research I found a nice paper from Abraham (https://projecteuclid.org/euclid.bams/1183525355) where the theorem is stated for $C^r$ manifolds.

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