Please Prove in Detail Explanation 10 points Let F R right a

Please Prove in Detail Explanation

(10 points) Let F: R right arrow R^m be continuous, and suppose for allepsilon > 0 and For all in R^n F(Bepsilon((x))subset Bepsilon(F(x)). Prove that F is uniformly continuous. Please Prove in Detail Explanation

Solution

F is given to be a continuous function from Rⁿ to R^m.

Consider a point x and the points around it satisfying d(x,a) < epsilon

As this is given to be contained in

B {F(x)} we have

All a in R^n satisfying d(x,a) < epsilon have images contained in neightbourhood of F(x)

i.e. for every d(x,a) < epsilon,d {f(x), f(a)} < epsilon1 for a suitable epsilon1 which depends on epsilon alone.

This implies that F is uniformly continuous.