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Is the following function: $$f(x)=\sum_{n=1}^\infty {\frac{\sin(nx^2)}{1+n^4}}$$ Has a continuous second derivative in R? (If it was in $[-a,a]$ or for $\sin(nx)$ it was easy, but I'm not sure what to do with the derivative of the inner function)

Thanks! (Even just an answer of yes/no will help me because I'm not sure if there is a problem with the question or me :P )

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Yes. If you know that the answer is positive when the function is restricted to $[-a,a]$, use that $f$ is $C^2$ on $\mathbb R$ if and only if $f$ is $C^2$ on every $(-a,a)$. –  Did Apr 28 '13 at 20:58
    
Why is it true? How can we show a uniform convergence in all R? (In [-a,a] we can say the if f(x)=Sum(Un(x)) than |Un(x)|<=a * 2n/(1+n^4) -> 0 yet we used the fact that |x|<=a ) –  ORBOT Inc. Apr 28 '13 at 21:02
    
Because being $C^2$ at a point $x$ is a local property, it does nor depend on the behaviour of the function outside of $(x-1,x+1)$, say. –  Did Apr 28 '13 at 21:04
    
Sounds good, thanks –  ORBOT Inc. Apr 28 '13 at 21:07

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