I am self studying analytic number theory and I could not deduce this statement given in Tom M Apostol.
Statement is - The function J takes every value exactly once in closure of fundamental region.
Two theorems have been proved earlier, both of whose proofs I understood completely ( They are very easy) . Theorem 1- If f is modular and not constant then f - c has same no. of zeroes and poles in closure of fundamental region where c is any constant belonging to $ C $ . In other words, f takes all values equally often in closure of fundamental region.
Theorem 2 - If f is modular and bounded then it is constant.
Now, deduction of statement to be proved - Using theorem 1 and 2 I can deduce that J takes all values equally oftenly as it is modular.
But can someone please explain how to prove that J takes all values exactly once.