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Does there exist a real differentiable function $f$ with following properties simultaneously ?

(1) $‎D_f‎\supseteq‎\mathbb{N}‎‎$,

(2) if put $f_n:=f(n)$, then the sequence $f_n$ is convergent as $n\longrightarrow‎ \infty$,

(3) if put $f^\prime_n:=f^\prime(n)$, then $f^\prime_n$ is not convergent as $n\longrightarrow‎ \infty$ .

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Yes. Try $x \mapsto \sin(\pi x)$. Then $f_n = 0$ and $f'_n = (-1)^n$.

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Take $$f (x)=\frac {\sin (x^2) }{x}$$ $$f_n\to 0$$

$$f'(x)=2\cos (x^2)-\frac {\sin (x^2)}{x^2} $$

$f'_n $ goes nowhere.

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  • $\begingroup$ How do you know $\cos n^2$ doesn't have a limit? $\endgroup$
    – zhw.
    Sep 12 '17 at 21:25
  • $\begingroup$ @zhw. we can use the subsequence $ cos (4n^2) $ $\endgroup$ Sep 13 '17 at 11:24

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