Let F : U ⊂ Rn → Rm be a differentiable function and p < m an integer. If k ≥ 1 is an integer, α ∈ [0, 1] and F ∈ Ck+(α), if we set Cp(F) = {x ∈ U | rank(Df(x)) ≤ p} then the Hausdorff measure of dimension (p + (n-p)/(k+α)) of F(Cp(F)) is zero.
The key result (Theorem 1) provides the existence of a holomorphic approximation map for some space of C∞-functions on an open subset of Rn. This leads to results about the existence of a continuous linear extension map from the space of the Whitney jets on a closed subset F of Rn into a space of holomorphic functions on an open subset D of Cn such that D ∩ Rn = RnF.