Could someone please check my work

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() > 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( ) f - (20 ) f1 # 8 > 10 - 201 ' ( = 2 A 7247 4ons 0 0 and f (c) 2 2 > 0 then f( ) > 0. Thus, there exists an a > 0 such that f(a) > > 0. Since |x - c) 0 and a neighborhood U of c such that f(a) > a for all & 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