<s>[INST] Explain the value of $(0,0)$ for $f(x,y)=1+x\frac{x^2-y^2}{x^2+y^2}$ 
Let $f:\mathbb R^2\to\mathbb R$ (where $\mathbb R$ is the real numbers) continuous in $\mathbb R^2$, such that 
  $$f(x,y)=1+x\frac{x^2-y^2}{x^2+y^2}$$ for $(x,y)\neq(0,0)$
Justify the value of $f(0,0)$.

The function is continuous in its domain, and $f(0,0)$ is defined by another equation. What else do I have to justify?
 [/INST] We have: $\left|x\cdot \dfrac{x^2-y^2}{x^2+y^2}\right| \le |x|\cdot \left|\dfrac{x^2}{x^2+y^2}+\dfrac{y^2}{x^2+y^2}\right| = |x| \implies \displaystyle \lim_{(x,y) \to (0,0)} f(x,y) = 1+0 = 1 \implies f(0,0) = 1$.
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