Homework 13: Continuity in Metric Spaces

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

1. Section 5.3, #3(a), (b).
2. Section 5.5, #15.
3. Let \((X,d)\) be a metric space and suppose that \(f:X\to \bR\) is continuous on \(X\). If \(D\subseteq X\) and \(f(x)=0\) for all \(x\in D\), prove that \(f(x)=0\) for all \(x\in \overline{D}\).
4. Let \((X,d)\) be a metric space and suppose that \(f:X\to \bR\) is continuous on \(X\). Let \(Z(f)\), the zero set of \(f\), be the set of all \(x\in X\) for which \(f(x)=0\). Prove that \(Z(f)\) is closed.
   Hint: The zero set of \(f\) can be identified as a pre-image. Use a theorem concerning pre-images of continuous functions.
5. Suppose that \(f:\mathbb{R}\to \mathbb{R}\) is continuous and that \(f(x) = 0\) for all \(x\in \mathbb{Q}\).
   1. Use Problem 4(a) of Homework 12 to prove that \(f(x)=0\) for all \(x\in \mathbb{R}\).
   2. Use Homework 8 and Problem 3 of this assignment to prove that \(f(x)=0\) for all \(x\in \mathbb{R}\).