I know some counterexamples where either $|f^\prime(x)|$ or $f(x)$ are not continuous. Now, I just can't come up with a counterexample where both $|f^\prime(x)|$ and $f(x)$ are continuous. I'd appreciate it if somebody could help.
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$\ln(x)$ on $0 \lt x \in \Bbb{R}$? Or are your limits in the extended reals? – Ben Dec 28 '21 at 14:59
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1@Ben I would say that $\lim \ln(x)$ exists (and is equal to $+\infty$...) – TheSilverDoe Dec 28 '21 at 14:59
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Just made an edit to the comment – Ben Dec 28 '21 at 15:00
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I am not familiar to the extended reals, but do we say that $\lim_{x\to a} f(x)$ exists when $\lim_{x\to a} f(x)= \infty$? – Hermis14 Dec 28 '21 at 15:03
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3$\sin(\ln(x))$ on $0<x$? – Gerd Dec 28 '21 at 15:03
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5and just modifying @Gerd's suggestion slightly, if you really want a ($C^{\infty}$) function defined on $\Bbb{R}$, then look at $f(x)=\sin(\ln(1+x^2))$. – peek-a-boo Dec 28 '21 at 15:08
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@peek-a-boo / Gerd Ah ja, that makes a lot of sense. Thanks for the quick help guys! – Mememaster696 Dec 28 '21 at 15:13
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1Also: https://math.stackexchange.com/q/3055363/42969, https://math.stackexchange.com/q/1607014/42969, https://math.stackexchange.com/q/2600562/42969. – Martin R Dec 28 '21 at 15:23