(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?