Real function/Continuous/Constant/Identity/Example

A constant function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto c,$

is continuous. For every given ${}\epsilon$, one can choose an arbitrary ${}\delta$, since

${}d(f(x),f(x'))=d(c,c)=0\leq \epsilon \,$

holds anyway.

The identity

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x,$

is also continuous. For every given ${}\epsilon$, one can take ${}\delta =\epsilon$, yielding the tautology: If

${}d(x,x')\leq \delta =\epsilon \,,$

then

${}d(f(x),f(x'))=d(x,x')\leq \epsilon \,$

holds.