# Properties of Continuous Functions

• Dec 6th 2012, 06:34 PM
Properties of Continuous Functions
If i have a continuous function "f" from the reals to the reals and a bounded subset of real numbers "B," is f(B) always bounded?

I'm trying to find some counter examples, but do counter examples even exist? (Headbang)
• Dec 6th 2012, 07:08 PM
GJA
Re: Properties of Continuous Functions
Hi Aqua,

If $B$ is bounded we can choose a natural number $N$ large enough so that $B\subseteq [-N, N].$ Since $[-N, N]$ is compact, $f([-N, N])$ is compact (because $f$ is continuous). Hence, $f([-N, N])$ is bounded. Can you see why $f([-N, N])$ being bounded implies $f(B)$ must be bounded?

Let me know if this gets things on the right track. Good luck!
• Dec 6th 2012, 08:00 PM