f-0-R-f-ux-x-u-f-x-f-x-

Question Number 3878 by 123456 last updated on 23/Dec/15

f : [0, + \infty) \to R

f (u x) = x^{u} f (x)

f (x) = ?

Commented by Rasheed Soomro last updated on 23/Dec/15

f (u x) = x^{u} f (x)

L e t x = 0

f (0) = 0^{u} f (0) = 0

x = 1

f (u x) = x^{u} f (x) \Rightarrow f (u) = f (1)

\Rightarrow f (1) = f (u) \Rightarrow u = 1 \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots (i)

S o, f (u x) = x^{u} f (x) \Rightarrow x^{u} = 1 \Rightarrow x = 1

F o r x = 1 a n d u = 1 f (x) m a y h a v e

a n y d e f i n i t i o n .

F o r e x a m p l e f (x) = x^{2}

f (u x) = x^{u} f (x) \Rightarrow {(u x)}^{2} = x^{u} (x^{2}) \Rightarrow 1 = 1 f o r x = 1, u = 1

Commented by prakash jain last updated on 23/Dec/15

f (x) = 0

I think f (x) = 0 is only solution .

Answered by RasheedSindhi last updated on 24/Dec/15

f (u x) = x^{u} f (x) \dots \dots \dots \dots . . (i)

x = 1 \Rightarrow f (u) = f (1) \Rightarrow u = 1

u = 1 \Rightarrow (i) w i l l b e c o m e

f (x) = x f (x)

x f (x) - f (x) = 0

f (x) (x - 1) = 0

f (x) = 0 ∣ x = 1