I am a math hobbyist/amateur studying complex analysis from Serge Lang's book Complex Analysis.

I need some help regarding Theorem 1.1 on Page 89

The theorem and is proof as given by Lang are as follows:

"Theorem 1.1 Let U be a connected open set, and let f be a holomorphic function on U. If f ' = 0 then f is constant.

Proof: Let be two points in U and suppose first that is a curve joining to so that

(a) = and (b) =

The function

t --> f( (t))

is differentiable, and by the chain rule, its derivative is

f ' ( (t)) '(t) = 0 .... (1)

Hence this function is constant, and therefore

f( ) = f( (a)) = f( (b)) = f( )

Next suppose that = { } is a path joining to , and let be the end point of , putting

By what we have just proved

f( ) = f( ) = f( ) = f( ) = ..... = f( )

thereby proving the theorem

QUESTIONS

(1) Lang does not describe the nature of the elements of U - but I am assuming that he is taking them as complex numbers? Is that right?

(2) At first sight - to someone familiar with elementary real analysis, the conclusion of the theorem that f ' = 0 implies f is constant does not seem surprising? Any comments?

(3) The statements in the proof:

"

The function

t --> f( (t))

is differentiable, and by the chain rule, its derivative is

f ' ( (t)) '(t) = 0 .... (1)

Hence this function is constant,"

seem to assume the proof since after assuming f ' = 0 Lang just states, "hence this function is constant".

I am lost at what is being proved - can someone please help?