Could someone please tell me if my proof is correct

er5 Please state all definitions and theorems that you "i" need: Definition 5.2.1 Let f: D + R and let 6 E D. We say that f is continuous at r.\" if - fic)| 6 Let f: D > R be continuous at c E D and suppose that f(c) > 0 . Prove that there exists an a > 0 and a neighborhood U of c such that f(m) > a for all :1: E U H D . fit) 2 Let f:D > IR be continuous at c E D and supposef(c) > 0. Leta = > 0 . ' Since f is continuous at c , then by definition 5.2.1, f(c) _ a V5>0 36>OsuchthathED,Iwc||f(:1:)f(c)| f(:1:) E V(f(c); 2) by the definition of neighborhood of f(c) - f(C) 2 Thus, there exists an oz > 0 such that f(z) > a > 0 . Since f(c) > 0 and > Othen f(m) > 0. / Since la: cl :1: E U(c; 6) by the definition of neighborhood of c, and since Vm E D, |:L' - cl O . then by Theorem 5.2.2(c) and definition 5.2.1, there exists an or > 0 and a neighborhood U of c such that f(a:) > a for all a: E U Fl D . Therefore, when f : D > R is continuous at c E D and f(c) > 0 , there exists an a > 0 and a neighborhood U ofc such that f(a:) > a for all to e U F'l D