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The question.

Let $f$ be a differentiable map $\mathbb R_+\to\mathbb R$ such that

  • $\forall x\geqslant0,\quad f'>0$

  • $\exists M>0,\quad \forall x\geqslant0,\quad f(x)\leqslant M$.

Does there exist $f$ verifying such conditions such that $f'(x)$ does not tend to $0$ when $x$ goes to infinity?

What I tried.

I think the answer is yes because of the following.

I tought about a function $f'$ doing something resembling to $x\mapsto \frac 1{1+x^2}$, but with little peaks like this:

enter image description here

This function would not have a limit when $x\to\infty$, and $f$ would still be bounded.

Final questions.

Does this idea works? Can you formalise it? Am I totally wrong here?

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    Duplicate of several duplicates... – 2017-01-22
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    f'(3/4) doesn't look positive – 2017-01-22
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    @AnalysisStudent0414 I know, but I had no better image. But I explain it in details in the question... – 2017-01-22
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    Also, you might want to purchase Hauchecorne, *Les Contre-Exemples en Mathématiques* – 2017-01-22
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    @LeGrandDODOM Actually I should have it somewhere ! :) I will look up there. – 2017-01-22

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