السلام عليكم أخي عمر
هذه محاولتي باعتبار { ........ N= { 0 , 1 , 2 , 3
الدوال التي تحقق هي f(x) = m*x حيث m belong to N

الإثبات :
(f(x^2+y^2)=x*f(x)+y*f(y

(put x = 0 ===>f(y^2)= y f(y
(and y =0 ====> f(x^2)=xf(x
===>
(f(x^2+y^2)=f(x^2)+f(y^2

x^2 belong to N
(f(x+y)= f(x)+f(y

(put y=0 ====> f(x)=f(x)+f(0
f(0)=0

(f(x+y-y)=f(x+y)+f(-y
(f(x)=f(x)+f(y)+f(-y
f(-y)=-f(y) ===> f is odd

put y = x+1

((f(x)+f(x+1)=f(x+x+1)=f(2x+1)=f((x+1)^2-f(x^2

=( f((x+1)^2 +f(-x^2
= ( x+1)f(x+1)-xf(x)
= ( x+1)(f(x)+f(1)- xf(x)
=(xf(x)+f(x)+xf(1)+f(1)-xf(x
=((xf(1)+(f(x)+f(1
=(xf(1)+f(x+1
====>
(f(x)+f(x+1)=xf(1)+f(x+1
(1)------- (f(x)=xf(1

now f : N---N
then we can let f(1)=m ,,, m belong to N
f(x)= m x من ( 1 )

chick this risult
f(x^2+y^2)= m(x^2+y^2)=m * x^2 + m * y^2
xf(x)+yf(y)=x*m x + y * m y=m * x^2 + m * y^2