---- > [!definition] Definition. ([[left-continuous, right-continuous]]) > Let $X$ be a [[poset|totally ordered set]] with [[order topology]] $\tau$, $Y$ a [[topological space]], and $f:X \to Y$ a function. > We say $f$ is **right-continuous** if it is [[continuous]] with respect to the [[lower limit topology, upper limit topology|lower limit topology]] $\tau^{-}$ on $X$. > We say $f$ is **left-continuous** if it is [[continuous]] with respect to the [[lower limit topology, upper limit topology|upper limit topology]] $\tau^{+}$ on $X$. > Note that $f$ is [[continuous]] with respect to the order topology $\tau$ on $X$ if and only if $f$ is left- and right-continuous, since $\tau=\tau^{+} \cap \tau^{-}$. ^definition > [!basicexample] > Consider the [[characteristic function|function]] $H:\mathbb{R} \to \mathbb{R}$ given by $\begin{align} H(x):=\begin{cases} 1 & x \geq 0 \\ 0 & x <0. \end{cases} \end{align}$ since for an [[interval]] $(a,b) \subset \mathbb{R}$ $H ^{-1} \big( (a,b) \big)=\begin{cases} \mathbb{R} & 0 \in (a,b) \text{ and } 1 \in (a,b) \\ [0, \infty) & 0 \not \in (a,b) \text{ and } 1 \in (a,b) \\ (-\infty, 0) & 0 \in (a,b) \text{ and } 1 \not \in (a,b) \\ \emptyset & 0 \not \in (a,b) \text{ and } 1 \not \in (a,b). \end{cases}$ we see that $H$ is right-continuous but not left-continuous (and hence not (order-)[[continuous]] either). ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```