# Thread: If f: R^n --> R^m is continuous everywhere and S is bounded, then f(S) is bounded

1. ## If f: R^n --> R^m is continuous everywhere and S is bounded, then f(S) is bounded

I have a general idea on how to go about it; if $\displaystyle \{x_{n}\}$ is a sequence in $\displaystyle S$, then it has a subsequence $\displaystyle \{x_{n_{k}}\}$that converges to some $\displaystyle x\in \mathbb{R}^{n}$, and since $\displaystyle f$ is continuous there $\displaystyle \{f(x_{n_{k}})\}$ is a convergent sequence in $\displaystyle f(S)$, which means that sequence is bounded. Now I am not sure if I can conclude that $\displaystyle f(S)$ is bounded. If I can't, how would I go about proving that if a function is continuous everywhere and a set S, subset of $\displaystyle \mathbb{R}^{n}$, is bounded, then $\displaystyle f(S)$ is bounded? Any help would be appreciated, thanks.

2. Originally Posted by Pinkk
If f: R^n --> R^m is continuous everywhere and S is bounded, then f(S) is bounded

I have a general idea on how to go about it; if $\displaystyle \{x_{n}\}$ is a sequence in $\displaystyle S$, then it has a subsequence $\displaystyle \{x_{n_{k}}\}$that converges to some $\displaystyle x\in \mathbb{R}^{n}$, and since $\displaystyle f$ is continuous there $\displaystyle \{f(x_{n_{k}})\}$ is a convergent sequence in $\displaystyle f(S)$, which means that sequence is bounded. Now I am not sure if I can conclude that $\displaystyle f(S)$ is bounded. If I can't, how would I go about proving that if a function is continuous everywhere and a set S, subset of $\displaystyle \mathbb{R}^{n}$, is bounded, then $\displaystyle f(S)$ is bounded? Any help would be appreciated, thanks.
If S is bounded then S is a subset of a compact set K, and the image f(S) of S under the continuous map f is a subset of the compact set f(K), and thus bounded.

3. Originally Posted by Pinkk
I have a general idea on how to go about it; if $\displaystyle \{x_{n}\}$ is a sequence in $\displaystyle S$, then it has a subsequence $\displaystyle \{x_{n_{k}}\}$that converges to some $\displaystyle x\in \mathbb{R}^{n}$, and since $\displaystyle f$ is continuous there $\displaystyle \{f(x_{n_{k}})\}$ is a convergent sequence in $\displaystyle f(S)$, which means that sequence is bounded. Now I am not sure if I can conclude that $\displaystyle f(S)$ is bounded. If I can't, how would I go about proving that if a function is continuous everywhere and a set S, subset of $\displaystyle \mathbb{R}^{n}$, is bounded, then $\displaystyle f(S)$ is bounded? Any help would be appreciated, thanks.
Assume that f(S) is not bounded.
Then there is a sequence f(x_n) in f(S) which has no convergent subsequence.
According to your argument above this is impossible. Therefore f(S) must be bounded.