Continuity of $f(x) = \left(\frac{2x}{1+x^2},\frac{1-x^2}{1+x^2}\right)$

real-analysis general-topology continuity stereographic-projections

Rational functions whose denominators are never zero are continuous and a function taking values in $\mathbb{R}^2$ is continuous if and only if it is coordinate-wise continuous. Both of these things imply your function is continuous.

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