Definition
2
[2DN] Let \(Aβ β\) and \(f:Aββ\) be a function; let \(x\in A\); \(f\) is called continuous at \(x\) if
\[ β \varepsilon {\gt}0,~ β πΏ {\gt} 0 , ~ β yβ A,~ |x-y|{\lt}πΏ βΉ |f(x)-f(y)|{\lt}\varepsilon ~ ~ . \]
\(f\) is called continuous if it is continuous in every point.
The set of all continuous functions \(f:Aββ\) is denoted with \(C(A)\); it is a vector space.