On pages 327 and 328 in R. Courant's book What Is Mathematics the global continuity of the function $x \to \frac{1}{1+x^{2}}$ is shown. But I don't understand that proof which runs essentially as follows:
On finding $$|f(x) - f(c)| = |x-c|\frac{|x+c|}{(1+x^{2})(1+c^{2})}$$ he says we can restrict $x$ to a fixed interval $|x| \leq M$ with $M$ an arbitrarily selected number. Then he says for $|x|, |c| \leq M$ we have $$|f(x) - f(c)| \leq |x-c|2M < \delta 2M\leq \epsilon$$.
Two things that I don't understand here are:
1) Why can we first choose $M$?
2) Why can we ignore the denominator?
What have I missed?
$|x+c| \leq |x| + |c| \leq M + M = 2M$, and $1+x^2 > 1$, $1 + c^2 >1$, and take $\delta = \dfrac{\epsilon}{2M}$, then conclusion follows.