Two hard problems - Help please!

Jan 2020
Hi Everyone

I'm new here and I would like to know if anyone can help me solving these two exercises:

- If f is a holomorphic function in this region: 0 <| z -z0 | < R (with R>0), show that if f is limited in that region ( |f(z)| ≤ M ), for every z inside that region, then z0 is a removable singularity.

-if g is an entire function and its codomain is included in the union of the real axis and the imaginary axis, what can you conclude?

Thank you