Find-all-continuous-functions-f-x-such-that-f-2x-1-f-x-for-all-real-x-

Question Number 6946 by 314159 last updated on 03/Aug/16

F i n d a l l c o n t i n u o u s f u n c t i o n s f (x) s u c h

t h a t f (2 x + 1) = f (x) f o r a l l r e a l x .

Commented by Yozzii last updated on 04/Aug/16

f (1) = f (0)

f^{'} (x) = 2 f^{'} (2 x + 1)

f (x) = c \in R \Rightarrow f (2 x + 1) = c = f (x) \forall x \in R

Commented by Rasheed Soomro last updated on 04/Aug/16

S a m e a n s w e r a s a b o v e b u t w i t h d i f f e r e n t p r o c e s s

S u p p o s e t h a t f (x) i s a l i n e a r p o l y n o m i a l a x + b

f (x) = a x + b

f (2 x + 1) = a (2 x + 1) + b = 2 a x + a + b

A s f (2 x + 1) = f (x)

S o 2 a x + a + b = a x + b

C o m p a r i n g c o e f f i c i e n t s :

a = 2 a \land a + b = b

a = 0, b = b

H e n c e f (x) = b

S a m e r e s u l t i f f (x) b e s u p p o s e d ? {a x}^{2} + b x + c