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

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 $n\longrightarrow‎ \infty$ .

Yes. Try $x \mapsto \sin(\pi x)$. Then $f_n = 0$ and $f'_n = (-1)^n$.
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.
• How do you know $\cos n^2$ doesn't have a limit?
• @zhw. we can use the subsequence $cos (4n^2)$ Sep 13 '17 at 11:24