> [!NOTE] > 1. If $\delta$ works in our definition then so does any $\delta' < \delta$. > 2. If $E$ is an open interval, $(a,b)$, then for any $c \in (a,b)$, if $\delta$ is sufficiently small then $|x-c| < \delta \implies x \in E$ > 3. If $E=[a,b]$ then at $c=a$, the condition is in fact $0\leq x-a < \delta$ > [!Remarks] > - An equivalent definition of [[Continuous Real Function]] is [[Continuous Function (Sequential Continuity Definition)]] (see [[Equivalence of Continuous and Sequentially Continuous Functions Over Euclidean Spaces]]). > - We say a function is *uniformly continuous* if our choice of $\delta$ does not depend on each individual point $c\in E$. > - Negating gives the definition of a [[Discontinuous Function (Epsilon-Delta Definition)]] at $c \in E$.