Could someone please check my work and let me know if it's correct

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(a) - f(c)| 0 such that If(a) | > a for all T EUnD. Suppose f : D - R is continuous at c E D and f(c) * 0. Let some real numbera > 0 . notjanut s to limit er hol . F.2 nobinfjab youmaria f ( c ) Let & = > 0 . Since f(c) # 0, then f(c) is a positive number. 2 Then by definition 5.2.1 for the continuity of f at c , VE > 0, 38 > 0 such that If(a) - f(c) | f(c) then f(a) > 0 when a E Un D sincef(c) is a positive number. If f ( ac) 0 . Since a > 0, then f(x) > 0. Since f(x) > 0 , then If(x) | > 0 . Therefore, there exists a neighborhood U of c and a real number a > 0 such that If(x) | > a for all x EUnD