if-f-R-R-is-continuous-and-f-x-y-f-x-y-1-find-f-x-2-proof-or-disproof-that-f-x-1-3-if-f-0-0-proof-or-disproof-that-f-x-x-

Question Number 652 by 123456 last updated on 19/Feb/15

i f f : R \to R i s c o n t i n u o u s a n d

f (x + y) = f (x) + y

1 . f i n d f (x)

2 . p r o o f o r d i s p r o o f t h a t f^{'} (x) = 1

3 . i f f (0) = 0 p r o o f o r d i s p r o o f t h a t f (x) = x

Answered by prakash jain last updated on 19/Feb/15

f (x + y) = f (x) + y

f (y) = f (0) + y

or f (x) = x + f (0) \dots . . (1)

f^{'} (x) = 1

f (0) = 0 \Rightarrow f (x) = x