Question about entire functions and Liouville's Theorem

Question

I have encountered a problem in complex analysis:

"Is there an entire function $f$ such that $f(0) = 0$ and $f(z) = 1$ whenever $|z|>1$? Justify your answer."

I know that it must have something to do with Liouville's Theorem, I just don't see it, can someone help me?

Answer 1

Yes, using Liouville's Theorem it is straightforward. The function is bounded by 1 since

1) |f(z)|=1 when |z|>=1. 2) |f(z)|<=1 when |z|<=1 ( Principle of Maximum)

So f(z) should be a constant, but it is not. So it does not exist