Rijen enzo
Zij ( X , d 1 ) {\displaystyle (X,d_{1})} en ( Y , d 2 ) {\displaystyle (Y,d_{2})} metrische ruimten.
Een functie f ( x ) : X → Y {\displaystyle f(x):X\rightarrow Y} is continu als:
∀ x , y ∈ X , ϵ > 0 {\displaystyle \forall x,y\in X,\epsilon >0} ∃ δ > 0 : d 1 ( x , y ) < δ ⇒ d 2 ( f ( x ) , f ( y ) ) < ϵ {\displaystyle \exists \delta >0:d_{1}(x,y)<\delta \Rightarrow d_{2}(f(x),f(y))<\epsilon } .
Een functie f ( x ) : X → Y {\displaystyle f(x):X\rightarrow Y} is uniform continu als:
∀ ϵ > 0 {\displaystyle \forall \epsilon >0} ∃ δ > 0 {\displaystyle \exists \delta >0} ∀ x , y ∈ X : d 1 ( x , y ) < δ ⇒ d 2 ( f ( x ) , f ( y ) ) < ϵ {\displaystyle \forall x,y\in X:d_{1}(x,y)<\delta \Rightarrow d_{2}(f(x),f(y))<\epsilon } .
Een verzameling F ⊆ C 0 ( X , Y ) {\displaystyle F\subseteq C^{0}(X,Y)} is equicontinu als:
∀ ϵ > 0 {\displaystyle \forall \epsilon >0} ∃ δ > 0 {\displaystyle \exists \delta >0} ∀ x , y ∈ X , f ∈ F : d 1 ( x , y ) < δ ⇒ d 2 ( f ( x ) , f ( y ) ) < ϵ {\displaystyle \forall x,y\in X,f\in F:d_{1}(x,y)<\delta \Rightarrow d_{2}(f(x),f(y))<\epsilon }
Een verzameling F ⊆ C 0 ( X , Y ) {\displaystyle F\subseteq C^{0}(X,Y)} is supercontinu[1] als:
∃ δ > 0 {\displaystyle \exists \delta >0} ∀ x , y ∈ X , ϵ > 0 , f ∈ F : d 1 ( x , y ) < δ ⇒ d 2 ( f ( x ) , f ( y ) ) < ϵ {\displaystyle \forall x,y\in X,\epsilon >0,f\in F:d_{1}(x,y)<\delta \Rightarrow d_{2}(f(x),f(y))<\epsilon }
en 
metrische ruimten.
is continu als:
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is uniform continu als:
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is equicontinu als:
is supercontinu[1] als:
