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Can the graph of a bounded function ever have an unbounded derivative?

I want to know if $f$ has bounded variation then its derivative is bounded. The converse is obvious. I think the answer is "yes". If the graph were to have an unbounded derivative, it would coincide with a vertical line.

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    Take $f(x)=\sqrt{\lvert x \rvert}$ for $x\in [-1, 1]$.2012-12-13

5 Answers 5