It is known that f (x) is a function defined on R which is not always zero, and for any x, y ∈ R, f (XY) = XF (y) + YF (x) The problem is if y = f (x) is an increasing function on [0, + ∞) and satisfies f (x) + F (x-1 / 2)

It is known that f (x) is a function defined on R which is not always zero, and for any x, y ∈ R, f (XY) = XF (y) + YF (x) The problem is if y = f (x) is an increasing function on [0, + ∞) and satisfies f (x) + F (x-1 / 2)

Let x = y = 0, then f (0) = 0
Let y = 1, then f (x) = f (1) x + F (x) and f (1) = 0
Let x = y = - 1, that is, f (1) = - 2F (- 1), so f (- 1) = 0
Let y = - 1 f (- x) = f (- 1) x-f (x), so f (x) is an odd function
Are there any other conditions?
Otherwise, we can only assume that there is x > 0 so that f (x) > 0