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Math Help - Proof concerning continuity

  1. #1
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    Proof concerning continuity

    Let f:[a,b] --> R be a continuous function such that [a,b] is a subset of [f(a), f(b)]. Prove that there exists x* in [a,b] such that f(x*)=x*
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  2. #2
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    Quote Originally Posted by paupsers View Post
    Let f:[a,b] --> R be a continuous function such that [a,b] is a subset of [f(a), f(b)]. Prove that there exists x* in [a,b] such that f(x*)=x*
    Let g(x)=f(x)-x and use the intermediate value theorem on g.
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  3. #3
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    Hm, I'm not following you... how will that help me prove f(x*)=x* for some x*?

    I understand the "geometry" of the problem, ie, the function is "taller" than it is "wide" so the line y=x must pass through the function. Just having a difficult time proving that.
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  4. #4
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    consider the function g(x)=f(x)-x, x \in [a,b]
    because [a,b] is a subset of [f(a), f(b)], then a \geq f(a) and f(b) \geq b, so g(a) = f(a) - a \leq 0 and g(b) = f(b) - b \geq 0.
    From the intermediate value theorem, there is c \in [a,b] such that g(c) = 0, that is g(c) = f(c) - c = 0
    hence f(c) = c
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  5. #5
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    Ah, thanks! That makes perfect sense!
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