Definition
15
[155] Let \(Aβ β\) and \(f:Aββ\) be a function; \(f\) is called uniformly continuous if
\[ β \varepsilon {\gt}0,~ β πΏ {\gt} 0 , ~ β x,yβ A,~ |x-y|{\lt}πΏ βΉ |f(x)-f(y)|{\lt}\varepsilon ~ ~ . \]
More in general, given \((X_ 1,d_ 1)\) and \((X_ 2,d_ 2)\) metric spaces, given the function \(f:X_ 1β X_ 2\), \(f\) is uniformly continuous if
\[ β \varepsilon {\gt}0,~ β πΏ {\gt} 0 , ~ β x,yβ X_ 1,~ d_ 1(x,y){\lt}πΏ βΉ d_ 2(f(x),f(y)){\lt}\varepsilon ~ ~ . \]