Could someone please tell me if my proof is correct

Please state all definitions and theorems that you will need: Definition 5.2.1 Let f: D - R and let c E D. We say that f is continuous at c if for every & > 0 there exists a d > 0 such that If(x) - f(c) | 0 and a neighborhood U of c such that If(x) | 0 and 3U of c such that I f(x) | Jo nA pue 0 n By Theorem 5.2.2(b), Since If(x) | > n Vn E N, then lim f(an) = f(c) . f(c) must equal zero so that If (a) - f(c)| = If(x) - 0| = If(x)| > n Since f is continuous and f(c)=0, then by definition 5.2.1, VE > 0 38 > 0 such that Va E D , |x - c) 0 such that there exists a U(c; 6) or rather, U of c such that I f (2) | n means f(x) is not within E of f(c) = 0, which means that there is at least one point, c where f(a) is discontinuous by Theorem 5.2.2(b), which is a contradiction of f being continuous at c E D by definition 5.2.1. Therefore, by contradiction, when f : D - R is continuous at c E D , there exists a M > 0 and a neighborhood U of c such that If (x) |