Exercise 11, § 6

Let $\overline{\mathbf{R}}$ carry the right topology (Chapter I, § 1, Exercise 2). A mapping $f$ of a topological space $X$ into $\overline{\mathbf{R}}$ is continuous in this topology if and only if $f$ has a relative minimum at every point of $X$. If $X = \mathbf{R}$ (with the usual topology), show that the set $f(X)$ is then countable. [Suppose that the result is false, and that the interval $[\alpha, \beta] \cap f(X)$ is uncountable; for each $y \in [\alpha, \beta]$, let $U(y) = \overline{f}^{-1}([y, +\infty])$; $U(y)$ is an open subset in $\mathbf{R}$. Let $(I_n)$ be the sequence (finite or infinite) of components of $U(\beta)$, and for each $n$ let $x_n$ be a point of $I_n$; for each $y \in [\alpha, \beta]$, let $g_n(y)$ (resp. $h_n(y)$) be the left-hand (resp. right-hand) end-point of the component of $U(y)$ which contains $x_n$. Show that for at least one value of $n$ one of the functions $g_n, h_n$ must take an uncountable infinity of distinct values in $[\alpha, \beta]$; now use Exercise 8 of § 5].