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

**Pinkk** · September 13th 2010, 07:20 PM

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

**Failure** · September 14th 2010, 04:04 AM

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 $\{x_{n}\}$ is a sequence in $S$ , then it has a subsequence $\{x_{n_{k}}\}$ that converges to some $x\in \mathbb{R}^{n}$ , and since $f$ is continuous there $\{f(x_{n_{k}})\}$ is a convergent sequence in $f(S)$ , which means that sequence is bounded. Now I am not sure if I can conclude that $f(S)$ 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 $\mathbb{R}^{n}$ , is bounded, then $f(S)$ 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.

**Iondor** · September 14th 2010, 06:15 AM

Originally Posted by Pinkk

I have a general idea on how to go about it; if $\{x_{n}\}$ is a sequence in $S$ , then it has a subsequence $\{x_{n_{k}}\}$ that converges to some $x\in \mathbb{R}^{n}$ , and since $f$ is continuous there $\{f(x_{n_{k}})\}$ is a convergent sequence in $f(S)$ , which means that sequence is bounded. Now I am not sure if I can conclude that $f(S)$ 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 $\mathbb{R}^{n}$ , is bounded, then $f(S)$ 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.

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