Is f(x,y) Linear if f(x,y)-f(0,0)=x*(d/dx)[f(x,y)]+y*(d/dy)[f(x,y)]?

[rman144]

I've been trying to prove that if the following statement holds for all (x,y)ER^2, f must be a linear function:

f(x,y)-f(0,0)=x*(d/dx)[f(x,y)]+y*(d/dy)[f(x,y)]

It seems to work for any function I plug in, but I'm unable to establish why this always works. Also, when I say (d/dx)[f(x,y)], I mean the derivative of f(x,y) with respect to x.

Thanks in advance for any help or ideas.

 

[arildno]

IF f(x,y) is a LINEAR function, then you know thay f(x,y)=Ax+By+C, with A, B and C being constants.

Thus, f(x,y)-f(0,0)=Ax+By.

Does that help?