If `f:RrarrR` is a continuous function satisfying `f(0)=1` and `f(2x)-f(x)=xAAxepsilonR` and `lim_(nrarroo)(f(x)-f(x/(2^(n))))=P(x)`. Then `P(x)` is

Options

A. a constant function
B. a linear function
C. a quadratic polynomial in `x`
D. a cubic polynomial in `x`

Answer

B

Explanation

Write the functional equation as
`f(x)-f(x/2)=x/2`
`impliesf(x/2)-f(x/4)=x/4`
`impliest(x/4)-f(x/8)=x/8`

`f(x/(2^(n-1)))-f(x/(2^(n)))=x/(2^(n))`
Adding up we obtain
`f(x)-t(x/(2^(n)))=x(1/2+1/4+ - - +1/(2^(n)))`
Hence `lim_(nrarroo)(f(x)-f(x/(2^(n))))=x`