Could someone please check my work and let me know if it's 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 o > 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 If() | 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) - 01 = If(x)| > n Since f is continuous and f(c)=0, then by definition 5.2.1, 8> 10 - 21 '( 3 xA 7247 4ons 0 0 such that there exists a U(c; 8) or rather, U of c such that If (2) | n means f(2) is not within E of f(c) = 0, which means that there is at least one point, c where f(x) 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) | S M for all & E Un D