Question

if f(x)+f(y)=f(x+y) prove that is an odd function

Answer 1

$f(x)+f(y)=f(x+y)$

put $x=y=0$ :
$f(0)+f(0)=f(0+0)$
$2f(0)=f(0)$
$f(0)=0$

put $y=-x$ : 
$f(x)+f(-x)=f(x-x)$
$f(x)+f(-x)=f(0)$
$f(x)+f(-x)=0$
$f(x)=-f(-x)$

which is exactly the definition of an odd function

Answer 2

f (x) + f (y) = f (x + y)    for all real values of x and y.

let x = 0, and y = 0.
     f(0) + f(0) = f (0+0) = f (0)
             => f (0) = 0      ---- (1)

let x = - y  then

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

   => f (-x) = - f ( x)      ,  as their sum is 0.          --- (2)

   =>  function f  is an odd function, as    (1) and (2)

   that is image wrt y axis is minus of its value, for an odd function.