Functions of algebra that deal with real number 
If the function $f$ satisfies the equation $f(x+y)=f(x)+f(y)$ for every pair of real numbers $x$ and $y$, what are the possible values of $f(0)$?
A.  Any real number
B.  Any positive real number
C.  $0$ and $1$ only
D.  $1$ only
E.  $0$ only

The answer for this problem is E. For the following problem to find the answer do you have to plug in 0 to prove the function?

Just to add to the collection above, if the vector space is $V$:
$$f(v)=f(v+0)=f(v)+f(0)\Longrightarrow f(0)=0$$
for any $v\in V$
