Could someone please tell me if my work 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 x 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 / # 8 > 10 - 21 ' 9 2A 7247 4ons 0 0 and f(C) 2 - > 0 then f(2 ) > 0 . Thus, there exists an a > 0 such that f(ac ) > a > 0. Since |x - c| 0 and a neighborhood U of c such that f(x) > a for all a E U. By Theorem 5.2.2(c) and definition 5.2.1, since f(x) E V whenever a E U , then there exists a neighborhood U of c