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 o > 0 such that If(x) - f(c)| 0. Prove that there exists an a > 0 and a neighborhood U of c such that f(x) > a for all a E Un D. Let f : D - R be continuous at c E D and suppose f(c) > 0. Let & = f(c) 2 > 0 . Since f is continuous at c , then by definition 5.2. 1, - > 1(2 ) f - (20 ) f 1 + 8 > 10 - 21 ' (1 3 2 A 32 47 yons 0 0 and - f(c) 2 > 0 then f(x) > 0 . Thus, there exists an o > 0 such that f(a) > a > 0. Since f is continuous and f(c)=0, then by definition 5.2. 1, 8 > 10 - 21 ' a 3 xA 7847 yons 0 0 such that there exists a U(c; 6) or rather, U of c such that If (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(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(a) |