Homework 12: Continuity

Instructions. Assignments should be stapled and written clearly and legibly.

1. Section 5.2, #4.
2. Let \(f:D\to \mathbb{R}\). Use the definition of continuity to prove that if \(c\) is an isolated point of \(D\), then \(f\) is continuous at \(c\).
3. Suppose \(f:\mathbb{R}\to \mathbb{R}\) is a function which satisfies \(|f(x)|\leq |x|\) for all \(x\in \mathbb{R}\). Using the definition of continuity, prove that \(f\) is continuous at \(0\).
4. Suppose that \(f,g,h\) are three functions which are defined on \((a,b)\) and continuous at \(c\in (a,b)\).
   1. Use the definition of continuity to prove that if \(f(c)\neq 0\), then there exists a neighborhood \(U\) of \(c\) such that \(f(x)\neq 0\) for every \(x\in U\).
   2. Prove that if \(g(c)\neq h(c)\), then there exists a neighborhood \(U\) of \(c\) such that \(g(x)\neq h(x)\) for every \(x\in U\). (Hint: consider the function \(p(x)=g(x)-h(x)\), and apply part (a)).
5. Let \(D\) be a subset of \(\mathbb{R}\) containing \(0\), and let \(f:D\to \mathbb{R}\) be bounded on \(D\) (i.e., \(f(D)\) is a bounded subset of \(\mathbb{R}\)). Define a new function \(g:D\to \mathbb{R}\) by \(g(x)=xf(x)\).
   1. Use the definition of continuity to prove that \(g\) is continuous at \(x=0\).
   2. Suppose \(c\neq 0\). Prove that \(g\) is continuous at \(c\) if and only if \(f\) is continuous at \(c\). (Hint: Use a theorem on continuity.)