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I'm given that $S$ consists of real numbers meaning there exists x$_n$ such that either lim(x$_n$)=+∞ or lim(x$_n$)=-∞ and I'm supposed to prove that there exists an unbounded continuous function on $S$

The best that I can find is f(x)=x and that if this exists in $S$ and $S$ is unbounded, then f(x) is unbounded for all x$_n$ in $S$ and continuous because f(x)=x is continuous. I found that solution online but it doesn't feel right. Is this enough for a proof? What can I do better?

Will Bothwell
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  • That solution is correct, although it could be structured more formally. A small issue: "$S$ consists of real numbers" does not imply "there exists ${x_n}\subset S$ such that $x_n\to \infty$ or $x_n\to -\infty$." I would probably rephrase this with a "such that" in the middle. – Michael L. Oct 18 '17 at 16:20
  • You could use $f(x) = |x|$, then you don't need to deal with a negative infinity as well. $f$ is continuous and there are $x_n \in S$ such that $f(x_n) \to \infty$. – copper.hat Oct 18 '17 at 16:28

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