Exercise 8.27

Following the hint, suppose that $f(z)\ne 0$ for all $z$. Then the continuous closed curves ${\gamma }_r(t)=f(r{e}^{\mathit{it}})$, $r\in [0,\infty )$, do not intersect the origin, so we can define the function $({\gamma }_r)$ for $r\in [0,\infty )$. We now show that this function would have the following three properties, which leads to a contradiction.

(a) $({\gamma }_0)=0$.

Since ${\gamma }_0(t)=f(0)$ is a differentiable function, we have $({\gamma }_0)={(2\pi )}^{-1}\int {\gamma }_0^{\prime }(t)∕{\gamma }_0(t)\mathit{dt}=0$.

(b) $({\gamma }_r)=n$ for all sufficiently large $r$.

By the assumption, we have

That is, for sufficiently large $r$ there is a $\delta >0$ such that

Let $\gamma (t)=c{r}^n{e}^{\mathit{int}}$, a differentiable closed curve which is nowhere 0. By the extension of Exercise 25 shown in Exercise 26, we have

(c) $({\gamma }_r)$ is a continuous function of $r$ on $[0,\infty )$.

Fix $r_0\in [0,\infty )$. Then the positive continuous function $|{\gamma }_{r_0}(t)|$ has a minimum value $𝜖>0$ on the compact set $[0,2\pi ]$ by Theorem 4.16. Since $f$ is continuous, by Theorem 4.19 it is uniformly continuous a compact neighborhood of the circle $\{r_0{e}^{\mathit{it}}\}$, $0\le t\le 2\pi$. Hence there is a $\delta >0$ such that if $|r-r_0|<\delta$ (limited to small positive values of $r$ in case $r_0=0$), we have

Hence, by the extension of Exercise 25 given in Exercise 26, we have

Note that a continuous, real-valued function on a connected set $X$ with only integer values must be constant. For then the inverse images of the open sets $(z-0.5,z+0.5)$ for the integers $z$ define a set of disjoint open sets which cover $X$, so by the connectedness of $X$ they must all be empty except one. Hence, since $[0,\infty )$ is a connected set, the properties (a), (b), and (c) of the function $({\gamma }_r)$ on $[0,\infty )$ lead to a contradiction.