Problem 1531 (difficulty: 9/10)

Let \(\displaystyle f:\CC\to\CC\) be a continuous function for which \(\displaystyle \displaystyle\lim_{z\to\infty}\frac{f(z)}z=1\) (i.e. \(\displaystyle \displaystyle\frac{f(z)}z\to1\) if \(\displaystyle |z|\to\infty\)). Show that the image of \(\displaystyle f\) is \(\displaystyle \CC\).

Give me another random problem!