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Math Help - Intermediate value theorem proof

  1. #1
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    Intermediate value theorem proof

    have a continuous function f:[0,1] -> [0,1]

    For some  x1, f(x1)>x1
    For some  x2, f(x2)>x2

    Using the Intermediate Value Theorem, show that there exists some x* such that f(x*)=x*

    I have attempted the problem:

    Make h(x)=f(x)-x. Then we are looking for the case when h(x)=0. So if f(0)=0 or f(1)=1, then we are done. So we assume neither case is true. Then we have that f(0)>0 because f must take on a value in the interval [0,1], and the same for f(1)<1. These two statements imply that h(0)>0 and h(1)<0. So then by the intermediate value theorem, we can say that h(x)=0 for some x [0,1] which means that  f(x)=x for some x [0,1]

    As I am pretty rusty on my proof skills I was wondering if somebody could check this for me.

    Thanks
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  2. #2
    Grand Panjandrum
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    Re: Intermediate value theorem proof

    Quote Originally Posted by whiteboard View Post
    have a continuous function f:[0,1] -> [0,1]

    For some  x1, f(x1){\color{red}<}x1
    For some  x2, f(x2)>x2

    Using the Intermediate Value Theorem, show that there exists some x* such that f(x*)=x*

    I have attempted the problem:

    Make h(x)=f(x)-x. Then we are looking for the case when h(x)=0. So if f(0)=0 or f(1)=1, then we are done. So we assume neither case is true. Then we have that f(0)>0 because f must take on a value in the interval [0,1], and the same for f(1)<1. These two statements imply that h(0)>0 and h(1)<0. So then by the intermediate value theorem, we can say that h(x)=0 for some x [0,1] which means that  f(x)=x for some x [0,1]

    As I am pretty rusty on my proof skills I was wondering if somebody could check this for me.

    Thanks
    The function h(x)=f(x)-x is continuous on an interval containing both x_1 and x_2 , with h(x_1)<0 and h(x_2)>0, hence by theIntermediate value theorem there exists a c between x_1 and x_2 such that h(c)=0 and so f(c)=c

    CB
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