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Math Help - Proof problem

  1. #1
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    Proof problem

    Hi,

    Can anybody offer any help/ideas with the following question:

    Suppose f is a differentiable function with | f '(x) | ≤ 1 for all x ε R . Show that

    | f(x) - f(y) | | x - y | for all x,y ε R .

    Will using the mean value theorem help with this proof?

    Thanks in advance.
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by jackiemoon View Post
    Hi,

    Can anybody offer any help/ideas with the following question:

    Suppose f is a differentiable function with | f '(x) | ≤ 1 for all x ε R . Show that

    | f(x) - f(y) | | x - y | for all x,y ε R .

    Will using the mean value theorem help with this proof?

    Thanks in advance.
    Of course oO

    Let x and y in R, x \neq y

    MVT says that there exists c between x and y such that :
    f'(c)=\frac{f(x)-f(y)}{x-y}

    Take absolute values of each side :
    \frac{|f(x)-f(y)|}{|x-y|}=|f'(c)|

    But we know that |f'(c)| \leq 1, for any c in R.

    Therefore...
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