If `f:R rarr R` is continuous function satisfying `f(0) =1` and `f(2x) -f(x) =x, AAx in R`, then `lim_(nrarroo) (f(x)-f((x)/(2^(n))))` is equal to

Options

A. `x`
B. `0`
C. `2x`
D. None of these

Answer

A

Explanation

Write the functional equation as
`f(x) -f(x//2)=x//2`
Also
`implies f(x//2)-f(x//4)=x//4`
Also `f(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)-f((x)/(2^(n)))=x((1)/(2)+(1)/(4)+(1)/(8)+....(1)/(2^(n)))`
`underset(nrarroo)lim(f(x)-f((x)/(2^(n))))=x`