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Math Help - Properties of Continuous Functions

  1. #1
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    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?
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  2. #2
    GJA
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    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!
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    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?
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