A constant function
is continuous. For every given ϵ {\displaystyle {}\epsilon } , one can choose an arbitrary δ {\displaystyle {}\delta } , since
holds anyway.
The identity
is also continuous. For every given ϵ {\displaystyle {}\epsilon } , one can take δ = ϵ {\displaystyle {}\delta =\epsilon } , yielding the tautology: If
then
holds.