---- > [!definition] Definition. ([[uniformly continuous]]) > Let $X$ and $Y$ be [[metric space|metric spaces]]. We say $f:X \to Y$ is **uniformly continuous** if for all $\varepsilon>0$, there exists $\delta>0$ such that $d(x_{1}, x_{2})<\delta$ implies $d\big( f(x_{1}), f(x_{2}) \big)<\varepsilon$ for all $x_{1},x_{2} \in X$. > Evidently, UC is much stronger than C in general. For [[compact]] [[metric space|spaces]] the notions coincide: [[continuous iff uniformly continuous on compact spaces]]. > [!basicexample] Example. > - Any [[Lipschitz continuous]] map is [[uniformly continuous]] (put $\delta:=\varepsilon / L$). ^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 > ```