- Suppose that \(f: \mathbb{R} \to \mathbb{R}\) is “continuous from the right,” that is,
\[ \lim_{x \to a^+} f(x) = f(a) \]
for each \(a \in \mathbb{R}\). Show that \(f\) is continuous when considered as a function from \(\mathbb{R}_\ell\) to \(\mathbb{R}\).
- Can you conjecture what functions \(f: \mathbb{R} \to \mathbb{R}\) are continuous when considered as maps from \(\mathbb{R}\) to \(\mathbb{R}_\ell\)? As maps from \(\mathbb{R}_\ell\) to \(\mathbb{R}_\ell\)? We shall return to this question in Chapter 3.
Solution
- Let \(V\subseteq\mathbb R\) be open and let \(a\in f^{-1}(V)\). Choose \(\varepsilon>0\) with \[ (f(a)-\varepsilon,f(a)+\varepsilon)\subseteq V. \] Right-continuity at \(a\) gives \(\delta>0\) such that \[ a\le x<a+\delta\implies |f(x)-f(a)|<\varepsilon. \] Therefore \[ [a,a+\delta)\subseteq f^{-1}(V). \] Since \([a,a+\delta)\) is a basic neighborhood of \(a\) in \(\mathbb R_\ell\), the preimage \(f^{-1}(V)\) is open in \(\mathbb R_\ell\). Hence \(f:\mathbb R_\ell\to\mathbb R\) is continuous.
In fact this condition is also necessary, so maps \(\mathbb R_\ell\to\mathbb R\) are exactly the ordinary right-continuous functions.
- A continuous map \(\mathbb R\to\mathbb R_\ell\) must be constant. Indeed each basic set \([c,d)\) in \(\mathbb R_\ell\) is both open and closed, so its inverse image is clopen in connected \(\mathbb R\); hence every such inverse image is either empty or all of \(\mathbb R\), forcing the image to be a singleton.
For maps \(f:\mathbb R_\ell\to\mathbb R_\ell\), continuity is equivalent to the following one-sided condition: for every \(a\in\mathbb R\) and every \(\varepsilon>0\), there exists \(\delta>0\) such that \[ f([a,a+\delta))\subseteq[f(a),f(a)+\varepsilon). \] Necessity follows by applying continuity to the basic target neighborhood \([f(a),f(a)+\varepsilon)\). Conversely, suppose the condition holds, and let \(a\in f^{-1}([c,d))\). With \(\varepsilon=d-f(a)\), the condition gives \(\delta>0\) with \(f([a,a+\delta))\subseteq[f(a),d)\subseteq[c,d)\), so \(f^{-1}([c,d))\) is open in \(\mathbb R_\ell\). Equivalently, such functions are right-continuous and locally nondecreasing from the right.