Section ext_cont.
Context {U : UniformSpace}.
Definition extension_cont (f g : R -> U) (a x : R) : U :=
match Rle_dec x a with
| left _ => f x
| right _ => g x
end.

Lemma extension_cont_continuous (f g : R -> U) (a : R) :
continuous f a -> continuous g a
-> f a = g a
-> continuous (extension_cont f g a) a.