Suppose we have the function $f(x, y) = (\frac{2x}{x^2 + y^2 + 1}, \frac{2y}{x^2 + y^2 + 1}, \frac{x^2 + y^2 - 1}{x^2 + y^2 + 1})$
where $f:\mathbb{R}^2 \to S^2 \setminus \{N\}$ where $N$ is the north pole of $S^2$
Can I prove that this is continuous without using topology, epsilon delta, or sequences?
I have read that "Clearly, it is a composition of continuous functions"
I just don't see which continuous functions I can use to compose the individual component functions.
I have also read that if you prove that the restriction to two different hemispheres is continuous, then the entire function is continuous.
I know that if $\sqrt{x^2 + y^2} > 1$, a point in the plane gets mapped to the upper hemisphere, if $\sqrt{x^2 + y^2} = 1$, to the equator, and if $\sqrt{x^2 + y^2} < 1$, to the lower hemisphere.
I prefer the easiest method. Thanks.