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Thread: proving that f' is bounded

  1. #1
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    proving that f' is bounded

    The problem I am trying to solve is
    $\displaystyle f:\Re\rightarrow\Re$ is a differentiable function and it has a continuous derivative. I need to prove that$\displaystyle f'$ is bounded on $\displaystyle [a,b]$ where $\displaystyle -\infty<a<b<\infty$.

    I have shown that if $\displaystyle f:X\rightarrow Y$ is continous on $\displaystyle X$ and if $\displaystyle A$ is a compact subset of $\displaystyle X$ that $\displaystyle f(A)$ is also compact. But have no clue if this will help of where to go from here.

    Thanks in advance
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  2. #2
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    Re: proving that f' is bounded

    Quote Originally Posted by klw289 View Post
    The problem I am trying to solve is
    $\displaystyle f:\Re\rightarrow\Re$ is a differentiable function and it has a continuous derivative. I need to prove that$\displaystyle f'$ is bounded on $\displaystyle [a,b]$ where $\displaystyle -\infty<a<b<\infty$.

    I have shown that if $\displaystyle f:X\rightarrow Y$ is continous on $\displaystyle X$ and if $\displaystyle A$ is a compact subset of $\displaystyle X$ that $\displaystyle f(A)$ is also compact. But have no clue if this will help of where to go from here.
    It tells you "it has a continuous derivative..
    Any continuous function is bounded on a compact set.
    Any interval $\displaystyle [a,b]$ is compact.
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