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For real $x$ and $y$ such that $(x,y)\ne (0,0)$, let:

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

...and let $f(0,0)=0$. Choose the correct response from the options below.

The function $f$ is continuous at the origin $(0,0)$.

The function $f$ has a removable discontinuity at the origin $(0,0)$.

The function $f$ has a non-removable discontinuity at the origin $(0,0)$.