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It looks to me that what you are really interested in is the Thom-Boardman stratification of the function space. For that I would recommend the well-written, Stable Mappings and There Their Singularities by Guillemin and Golubitsky (in the Springer GTM series).

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It looks to me that what you are really interested in is the Thom-Boardman stratification of the function space. For that I would recommend the well-written, Stable Mappings and There Singularities by Guillemin and Golubitsky (in the Springer GTM series).