Let f(x,y) = x^2 y - xy = x(x-1)y be a polynomial in k[x,y]. I am looking for the singular subset of this function. Taking the partials, we obtain f_x = 2xy - y f_y = x^2 - x. In order to find the singular subset, both partials (with respect to x and with respect to y) must vanish. So we obtain that f_x = 2xy - y = 0 which implies x = 1/2 or y = 0, while f_y = x^2 - x = 0 implies x = 0 or x =1. Drawing a picture of f, it is clear that the two points (0,0) and (1,0) are singular points, but what does x = 1/2 tell us? Is this point supposed to be a singular point as well?