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If the derivative of a function is lipschitz,,,does it mean that the function itself is also lipschitz? Any proof for that?

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  • $\begingroup$ A consequence of the mean value theorem is that the best Lipschitz constant for a derivable function on an interval is equal to the uniform norm of its derivative. $\endgroup$
    – Pietro Majer
    Commented Sep 21, 2012 at 7:56

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No. $f(x)=x^2$ on the whole real line is not Lipschitz, but the derivative $f'(x)=2x$ is. However, if the function is defined on a bounded interval, then the statement is true. Indeed, if $f'$ is Lipschitz, then it is continuous and thus bounded, and a function with bounded derivative is Lipschitz.

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