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I was trying to check the validity of the following:

If $f:\mathbb R\rightarrow\mathbb R$ and its derivative $f'$ are unbounded, then $f$ is not uniformly continuous on $\mathbb R$.

To me,the statement appears to be valid at first sight, but I am not sure. I know that boundedness of $f'$ is sufficient to prove that $f$ is uniformly continuous in its domain. Further, if $f'$ is unbounded, then $f$ may or may not be uniformly continuous. I thought of $f(x)=x^2$ with its derivative $f'(x)=2x$, both of which are unbounded on $\mathbb R$ and supports the statement, but I am not sure whether a counterexample exists. Please, correct me if I am wrong to conclude anything from this statement.

Nitin Uniyal
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1 Answers1

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Then answer is no, the statement is not true. Take for example the cube root $x \mapsto x^\frac{1}{3}$. This map is uniformly continuous on $\mathbb{R}$. But its derivative and itself are unbounded.

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Oles Wohnzimmer
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