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Math Help - Bolzano's Theorem and continuity?

  1. #1
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    Bolzano's Theorem and continuity?

    Ok so the question we were given was:

    Suppose that f is continuous with |f(x)| \leqslant 1 for all x \in R (R being the real numbers). Show that there exists some c \in R such that f(c) = c. Hint: apply Bolzano's theorem on a suitable interval.

    I tried a few different things originally, but I just couldn't get it out, and handed up my homework. I then found out that the solution he gave was:

    "Being the difference of two continuous functions, g(x) = f(x) - x is continuous with
    g(2) = f(2) - 2 \leqslant 1 - 2 < 0
    g(-2) = f(-2) + 2 \geqslant -1 + 2 > 0
    According to Bolzano’s theorem then, some c \in (-2, 2) exists such that g(c) = 0."

    I don't really understand how this proves that c \in R such that f(c) = c, though? Also, why pick (-2, 2) as the interval for g? If someone could explain it to me please, that'd be very handy!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Conn View Post
    I don't really understand how this proves that c \in R such that f(c) = c, though?

    g(c)=f(c)-c=0\Rightarrow f(c)=c


    Also, why pick (-2, 2) as the interval for g?

    This assures g ( a ) < 0 and g ( b ) > 0 .
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