Check continuity of a function $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ given by formula:
$f(x,y,z)=\begin{cases} \frac{xz + yz}{x^2+y^2+z^2} \text{ for }(x,y,z)\neq (0,0,0) \\ 0 \text{ for } (x,y,z)=(0,0,0) \end{cases}$
I don't know how to approach this. I tried to find two examples of sequences $(x_n,y_n,z_n)$ convergent to zero on all coordinates for which $f$ has different limits but it failed. I think $f$ can be continuous.