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Thread: one-one problem....

  1. #1
    Junior Member
    Jan 2009

    one-one problem....

    Q:let f from R to R be continuous function with mod(f(x)-f(y))>=mod(x-y) for every x,y belongs to R.Is f is one-one?.show that there can't exist 3 points a,b,c belongs to R with a<b<c s.t f(a)<f(c)<f(b).
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  2. #2
    Junior Member woof's Avatar
    Apr 2009
    Take $\displaystyle x \not= y $, but $\displaystyle f(x)=f(y)$, that is, you are assuming f is not 1-1. Then the inequality will not hold and the assumption (not 1-1) is false.

    If the function is 1-1, then it will either be increasing, or decreasing. If mod is absolute value, then you will have cases: $\displaystyle f(a)>f(b), f(a)=f(b), f(a)<f(b), $ Case 1 automatically proves it, case 2 can't happen from argument above. In case three, assume there is a c where f(c) is in the middle, and use the inequality to show that this can't happen.
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