I'm trying to solve the following problem:
Determine which $n\in\mathbb{N}$ make the following function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ continuous at the origin:
$ f(x,y,z)=\left\{ \begin{array}{cr} \frac{(\cos^2\left(|x|+|y|\right)-1)\sin(y^2+z^2)}{(x^2+y^2+z^2)^{n/2}} & \text{if } (x,y,z) \neq 0\\ 0 & \text{if } (x,y,z) = 0\\ \end{array} \right. $
It's obvious that $f$ is continuous at the origin if $n\leq 2$, since:
$\left|\frac{(\cos^2\left(|x|+|y|\right)-1)\sin(y^2+z^2)}{(x^2+y^2+z^2)^{n/2}}\right|\leq\frac{|\cos^2\left(|x|+|y|\right)-1|\,|y^2+z^2|}{|x^2+y^2+z^2|}\leq\frac{|\cos^2\left(|x|+|y|\right)-1|\,|y^2+z^2|}{|y^2+z^2|}=|\cos^2\left(|x|+|y|\right)-1|\rightarrow 0$
I'm almost sure the function isn't continuous if $n> 2$ but I can't seem to find a way to prove it.