Let `f : R rarr R` be a function such that `f(x+y) = f(x) + f(y), AA x, y in R`. If f(x) is differentiable at x = 0, then

Options

A. f(x) is continuous for all `x in R`
B. f'(x) is constant for all `x in R`
C. f(x) is differentiable for all `x in R`
D. f(x) is differentiable only in a finite interval containing zero

Answer

D

Explanation

We have
`f(x+y)=f(x)+f(y)"for all "x,y in R`
`Rightarrow f(0)=f(0)+f(0)["Replacing x and y both by zero"]`
`Rightarrow f(0)=0`
Now,
`f'(0)=underset(h to 0)lim (f(0+h)-f(0))/(h) Rightarrow f'(0) underset(h to 0)lim (f(h))/(h)`
`therefore f'(x)=underset(h to 0)lim (f(x+h)-f(x))/(h) underset(h to 0)lim (f(x)+f(h)-f(x))/(h)`
`Rightarrow f'(x)=underset(h to 0)lim (f(h))/(h)=f'(0)`
`Rightarrow f(x)=xf'(0)+C`
But, f(0)=0
`therefore C=0`
Hence, f(x) =xf'(0) for all `x in R`
Clearly, f(x) is everywhere continuous and differential and f'(x) is constant for all `x in R`
Alter, we have
`f(x+y)=f(x)+f(y)"for all"x,y in R`
`Rightarrow f(x)=xf(1)"for all"x,y in R`
Clearly, f(x) is everywhere continuous and differentiable such that f'(x)=f(1)=constant for all `x inR`