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

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

Can $f(0,0)$ be defined in such a way as to make $f$ continuous at the origin $(0,0)$?

No. The function does not have a removable discontinuity at the origin because $$\lim_{(x,y)\to(0,0)} f(x,y)$$ does not exist.

Yes. Define $f(0,0)=0$.

Yes. Define $f(0,0)=1$.