Space 1

Let m be a non zero element of N, f be a partial function from Rm to R, k be an element of N, and Z be a set. We say that f is continuously differentiable up to order of k and Z if and only if (Def. 1) (i) Z ⊆ dom f , and (ii) f is partial differentiable up to order k and Z, and (iii) for every non empty finite sequence I of elements of N such that len I ¬ k and rng I ⊆ Segm holds f Z is continuous on Z. Now we state the propositions: (1) Let us consider a non zero element m of N, a set Z, a non empty finite sequence I of elements of N, and a partial function f from Rm to R. Suppose f is partially differentiable on Z w.r.t. I. Then dom(f Z) = Z. This work was supported by JSPS KAKENHI 22300285.