> [!NOTE] Definition (Real Interval) > A non-empty set of [[Real numbers|reals]] $I \subset\mathbb{R}$ is an *interval* if whenever $x,y \in I$ and $x<y$ then $[x,y] \subset I$ (where $[a,b]=\{ x \in \mathbb{R} \mid a\leq x \leq b \}$). # Properties By [[Types of Real Intervals]], an interval can be classified as open or closed (left- or right- or both); bounded or unbounded (above or below or both); or degenerate (containing a single real number). > [!NOTE] Lemma (Continuous Image of an Interval is an Interval) > Let $I$ be an interval and $f: I \to R$ be [[Continuous Real Function|continuous]]. Then $f(I)$ is an interval. >See [[Continuous Image of an Interval is an Interval|proof]]. > [!NOTE] Definition (Length of bounded interval) > If $I$ is a bounded interval (one of the intervals $[a,b],(a,b),[a,b),(a,b]$), we define then length of $I$, denoted $|I|=\sup I -\inf I = b-a.$ > > [!NOTE] Definition (Partition) > A [[Finite Partition of Closed Real Interval|partition]] of a bounded interval $I$ is a finite set $P$ of bounded intervals contained in $I,$ such that every $x \in I$ lies in exactly one of the bounded intervals $J\in P.$ >See application: [[Riemann Integration]].