In Guillemin and Pollack, the Stability Theorem states that local diffeomorphisms, immersions, submersions, maps transversal to any specified submanifold ZY , embeddings, and diffeomorphisms of a compact manifold X into a manifold Y are stable classes, where a property is stable if whenever f0:XY possess the property and ft:XY is a homotopy of f0 , then for some >0 , each ft with t< also possesses this property. The collection of maps that possess a particular stable property is a stable class of maps. The proof of this theorem is given in Guillemin and Pollack, but I'm currently trying to find a proof for the following: Suppose fs:XY is a family of smooth maps, indexed by a parameter s that varies over a subset S in some Euclidean space. We say that {fs} is a smooth family of mappings if the map F:XSY , defined by F(x,s)=fs(x) is smooth. Check that the Stability Theorem generalizes immediately to the following: if f0 belongs to any of the classes listed, then there exists an >0 such that fs belongs to the same class if s0s< .