let-f-U-R-n-R-p-be-an-application-where-U-is-an-open-set-prove-that-x-U-h-R-n-such-as-x-h-U-

Question Number 71548 by Cmr 237 last updated on 17/Oct/19

let f : U \subset R^{n} \to R^{p} be an application

where U is an open set

prove that \forall x \in U, \exists h \in R^{n} such as

x + h \in U

Answered by mind is power last updated on 17/Oct/19

U is open \Rightarrow \exists ε > 0 such that B (x, ε) \subset U

we have B (x, ε) is a convex

(1 + \frac{ε}{2}) x \in B (x, ε) \subset U \Rightarrow (1 + \frac{ε}{2}) x \in U