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**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