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Choose the correct statement concerning the function $f$ defined for all real $x$ and $y$ such that $(x,y)\ne (0,0)$ by:

$$f(x,y)=\cfrac{x^4-2y^2}{x^4+y^2}$$

...and $f(0,0)=0$.

The function $f$ is continuous at every point of the Cartesian plane.

The function $f$ is continuous except at the origin, where it has a removable discontinuity.

The function $f$ is continuous except at the origin, where it has a non-removable discontinuity.