Section ext_C0.
Context {V : NormedModule R_AbsRing}.
Definition extension_C0 (f : R -> V) (a b : Rbar) (x : R) : V :=
match Rbar_le_dec a x with
| left _ => match Rbar_le_dec x b with
| left _ => f x
| right _ => f (real b)
end
| right _ => f (real a)
end.

Lemma extension_C0_continuous (f : R -> V) (a b : Rbar) :
Rbar_le a b
-> (forall x : R, Rbar_le a x -> Rbar_le x b -> continuous f x)
-> forall x, continuous (extension_C0 f a b) x.