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

Question

I am looking at a proof that shows $S^1$ is a topological manifold, where the author defines this function: $$ f_1^{-1}: \mathbb R \to S^1 \setminus \{(0,-1)\} $$ by $$ f_1^{-1}(x) = \biggl( \frac{2x}{1+x^2},\, \frac{1-x^2}{1+x^2} \biggr). $$ (Here $S^1$ is defined as the unit circle on $\mathbb R^2$ equipped with the subspace topology.) The author claims that the function is "evidently continuous when viewed as a function to $\mathbb R^2$". Can anyone provide an explanation to this claim? I know it is probably true if I just play around with the $\varepsilon$-$\delta$ argument, but is there a more obvious explanation?

Answer 1

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.