a function f(x,y) Any suggestion of a function f(x,y) such that the partial derivative with respect to the variable x is ax and with respect to the variable y is bx?
No, the system isn't inconsistent; just do the math: Integrating the first condition gives that f(x,y)=(a/2)x^2 + c(y). From this one obtains an, apparent, contradiction, since the derivative of this expression wrt y is a function only of y and can't be equal to bx; unless b=0, which means that c(y)=c(0)=c, is a constant.
If one started out by integrating wrt y, one would've found f(x,y)=bxy + d(x); the derivative of this expression wrt x would then give by + d'(x)=ax=>d'(x)=ax-by, which, once more, is consistent only with b=0.
So the only solution is f(x,y)=(a/2)x^2+c. Standard, elementary, exercise in undergraduate calculus. A mystery what it's doing on such a forum...
Try some function whose mixed partials are not continuous, cf Clairaut's Thm.
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