Let f(x) be a deriviable function at x=0 and f {(x+y) / k} = f{ (x) + f(y) } / k for ( k belonging to R and k ≠ 0,2) Then show that f(x) is either zero or odd function.

Dear Sanchit Gupta

Given

f {(x+y) / k } = {f  (x) + f(y) } / k

Put x=y=0

f(0)=(2/k)f(0)

f(0) [k-2]/k =0

but can not be 2

so f(0) =0

now put y=-x

f(0) =1/k{f(x)+f(-x)}

or   f(-x)=-f(x)

f(x) is a odd function