Assume the function f(x, y) is continuous on R2 and that for point (a, b) in R2, we have f(a, b) > 0. Prove that there exists a real number r > 0 such that for all x, y, |x−a| < r,&|y−b| < r ==> f(x, y) > 0. (this is relevant to the beginning of the proof of IFT.) Attempt: So we know f(x,y) is continuous on R2 and at (a,b), f(a,b) >0 ==> there exists an r1 > 0 s.t |x-a|< r1 and |y-b|< r1 ==> f(a,b) > 0. So from what I got to now which was essentially just a restatement of my assumptions, I have to somehow show that this is the case not for one point, but for all (x,y). My issue is that it does not say if the function is C1 so I'm kind of stuck when it comes to trying to apply the implicit function theorem which is what the objective of this question is.