# Properties of Continuous Functions

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• Dec 6th 2012, 06:34 PM
Aquameatwad
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 \$\displaystyle B\$ is bounded we can choose a natural number \$\displaystyle N\$ large enough so that \$\displaystyle B\subseteq [-N, N].\$ Since \$\displaystyle [-N, N]\$ is compact, \$\displaystyle f([-N, N])\$ is compact (because \$\displaystyle f\$ is continuous). Hence, \$\displaystyle f([-N, N])\$ is bounded. Can you see why \$\displaystyle f([-N, N])\$ being bounded implies \$\displaystyle f(B)\$ must be bounded?

Let me know if this gets things on the right track. Good luck!
• Dec 6th 2012, 08:00 PM
Aquameatwad
Re: Properties of Continuous Functions
I know by the preservation of compact sets if f:R->R is continuous on R and if B is a subset of R is compact then f(B) is compact.

If B were to be closed i know f(B) is not always closed (ex if g(x)=1/(1+x^2), B=[0, inf ) then g(B)=(0,1] which is not closed).

So your saying since B is inside the compact interval [-N,N], then f(B) must be bounded as well?