Why does the sign of ax+by+c change, for the coordinates of a point that is not on the line? So, all the points (x,y) satisfying ax+by+c = 0, lie on the staright line. But if a point is above or below that line, the sign of the function f(x,y) = ax+by+c, changes, being negative or positive, if the point lies on one of the semiplanes determined by the line. How can you prove that for all the points that are in a such semiplane, the sign of that function changes accordingly.
For example, if I have the straigt line x-y = 0:
For (1,1), 1 - 1  = 0 (on the straight line)
For (1,3), 1 - 3 = -2 (above the line)
For (3,-2), 3 + 2 = 5 (below the line)
So, for all the points that lie above the line the sign is -. For all the points that lie below, the sign is +. How can you prove this for the general case ax+by+ c = 0 ?

By continuity, if there are two points $(x,y)$ and $(z,w)$ such that $f$ is positive at one and negative at the other, say $f(x,y)>0,\hspace{3mm} f(z,w)<0$, then there must exist a point on the line segment in between them where the function is zero (intermediate value theorem). But the only zeroes of $$f(x,y)=ax+by+c$$ are on one line, so two points on the same side of the line must always have the same sign (when they are put into the function).
