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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.

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Try along the line $x=y=z$ and also along the $z$-axis. – David Mitra May 24 '12 at 21:07
up vote 0 down vote accepted

HINT Approach $(0,0,0)$ along $y = k_1 x$ and $z = k_2x$ for different $k_1,k_2$.

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