Show $z\mapsto\frac{\exp(\exp(z+z^2+z^3))}{1+z\overline{z}}$ is continuous on $\mathbb{C}$ and takes on the value $3$.

Question

Show that $f:\mathbb{C}\rightarrow\mathbb{C}, z\mapsto\frac{\exp(\exp(z+z^2+z^3))}{1+z\overline{z}}$ is continuous on $\mathbb{C}$ and takes on the value of $3$.

Answer 1

For $\varphi(x)=\dfrac{\exp(\exp(x+x^{2}+x^{3}))}{1+x^{2}}$, $x\in{\bf{R}}$, it is not hard to see that $\lim_{x\rightarrow\infty}\varphi(x)=\infty$, $\lim_{x\rightarrow-\infty}\varphi(x)=0$, so by Intermediate Value Theorem $\varphi(x_{0})=3$ for some $x_{0}\in{\bf{R}}$, then $f(x_{0})=3$ as well.

The continuity of $f$ can be argued in a successive way: The map $z\rightarrow 1+z\overline{z}=1+|z|^{2}$ is continuous and bounded away from zero, then the map $z\rightarrow\dfrac{1}{1+z\overline{z}}$ is continuous either.

The map $z\rightarrow z+z^{2}+z^{3}$ is continuous, so is $u\rightarrow\exp(u)$, then the composite function $z\rightarrow\exp(z+z^{2}+z^{3})$ is continuous either, and do the composition with $\exp$ again, one gets the continuity of the numerator of $f$.

Now the multiplication of two continuous functions is still continuous, so the continuity of $f$ is obtained.