Definition continuous_on (D : T -> Prop) (f : T -> U) :=
  forall x, D x -> filterlim f (within D (locally x)) (locally (f x)).Definition continuous (f : T -> U) (x : T) :=
  filterlim f (locally x) (locally (f x)).

Lemma continuous_mult {U : UniformSpace} {K : AbsRing} (f g : U -> K) (x : U) : continuous f x -> continuous g x -> continuous (fun y => mult (f y) (g y)) x.