As the book says, you only have to apply the definition of continuity, choose .
please see here(p.174-175 Elementary real and complex analysis
By Georgiĭ Evgenʹevich Shilov):
Elementary real and complex analysis - Google Books
How does modulus (H(z))<1/2
it doesn't make sense to me!
P.S. There is a lot of people in this forum who honestly try to help. So, I consider a lack of courtesy on your part to decide the way we have to answer.
It is not our job to teach you. Before you try to run, learn to walk. Before you try to walk, learn to crawl. Otherwise, a lot of people, including you, are going to get very frustrated.
I will second and underline what Mr. F posted above.
I will add a suggestion. Forget Google Books.
Get yourself an actual textbook suitable for you lack of background in these matters. I suggest that you look at an elementary text such as Ken Rossís Elementary Analysis.
An author such as Shilov is a challenge for even advanced students.
Elliott Mendelsonís Number Systems and the Foundations of Analysis in the Dover edition is quite inexpensive.