Originally Posted by
knguyen2005 When Im trying to do this question , I got stuck
Prove that f:A-->R^m is continuous ata if and only if each f^i is continuous at a for i=1,2,...,m
This is my attempt:
We have to prove 2 sides(=> and <=)
(=>) Let a belongs to A then f(a) is in R^m
Since f is continuous at a, we have:
Given e>0, I can find d>0 such that 0<|x-a|< d implies |f(x)-f(a)|<e
But each f^i is a scalar field component of f(x) = {(f^1(x),...,f^m(x)}
My question is can I put f(x) = {(f^1(x),...,f^m(x)} into |f(x)-f(a)|to yield the results?
Is there any better ideas to do this question
Thank you so much