Exercise 16, § 6

Let E be a locally connected topological space, $f$ a numerical function defined in E. Suppose that: 1° for every connected subset A of E, $f(A)$ is a connected subset of $\overline{\mathbf{R}}$ (in other words, an interval); 2° there exists in $\mathbf{R}$ a everywhere dense set D such that for every $y \in D, f^{-1}(y)$ is closed in E. Prove that under these conditions $f$ is continuous. (For every $y_0 \in D$, prove that the set of $x \in E$ such that $f(x) > y_0$ is open, by considering the connected components of the complement of $f^{-1}(y_0)$). Prove that the property does not extend to spaces E which are not locally connected (consider the characteristic function of the connected component of an open set).