Section UnifCont_N.
Context {K : AbsRing} {V : NormedModule K}.

Lemma unifcont_normed_1d (f : R -> V) a b :
(forall x, a <= x <= b -> continuous f x) ->
forall eps : posreal, {delta : posreal | forall x y,
a <= x <= b -> a <= y <= b -> ball x delta y -> ball_norm (f x) eps (f y)}.