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> Can the function $f(x,y)=\frac{x^2}{x^2+y^2}$ be made continuous at $(0,0)$ by defining $f(0,0)$ to be a specific value?
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I say yes. I solved it this way:

$y=x \Rightarrow \frac{x^2}{x^2+x^2} \Rightarrow \frac{x^2}{2x^2} \Rightarrow \frac{1}{2}$

Is my thinking on this correct?