continuity of composite function

Let f:R-------R be a continuous function defined by f(x+y)=f(x) +f(y) for all x,y belongs to R. if the function is continuous at x=0 then it is continuous at all x.

Sir in solution of this question they do like that.

For f (0) we substitute x=y=0 in the given functional equation relation, to get f (0) ==0 .sir how can we put x=0 and y=0 at our own will. How can we say that x and y are independent.

Again sir if f(x) is a polynomial function (say) then f(x) =x which clearly shows that x and y are not independent, as if i put x=1 then y=f(x) =1 shows that y depends upon x.