intros f g a x a0 q cf eps ep.
destruct (cf eps ep) as [a' [a'p Pa']].
exists (Rmin a a'); split.
- unfold Rmin; destruct (Rle_dec a a').
+ assumption.
+ assumption.
- intros y cy; rewrite <- !q.
+ apply Pa'.
split;[| apply Rlt_le_trans with (Rmin a a');[ | apply Rmin_r]];tauto.
+ rewrite Rdist_eq; assumption.
+ apply Rlt_le_trans with (Rmin a a');[ | apply Rmin_l]; tauto.
Qed.

Lemma continuity_pt_comp :
forall f1 f2 (x:R),
continuity_pt f1 x -> continuity_pt f2 (f1 x) -> continuity_pt (f2 o f1) x.