Require Export Continuity.

(* Specifications: *)
Lemma continuous_function_preserves_filter_limits:
forall (X Y:TopologicalSpace) (f:point_set X -> point_set Y)
(x:point_set X) (F:Filter (point_set X)),
filter_limit F x -> continuous_at f x ->
filter_limit (filter_direct_image f F) (f x).
Lemma func_preserving_filter_limits_is_continuous:
forall (X Y:TopologicalSpace) (f:point_set X -> point_set Y)
(x:point_set X),
(forall F:Filter (point_set X), filter_limit F x ->
filter_limit (filter_direct_image f F) (f x)) ->
continuous_at f x.
Lemma metric_space_fun_continuity_converse: forall (X Y:TopologicalSpace)
(f:point_set X->point_set Y) (x:point_set X)
(dX:point_set X -> point_set X -> R)
(dY:point_set Y -> point_set Y -> R),
metrizes X dX -> metrizes Y dY ->
continuous_at f x -> forall eps:R, eps > 0 ->
exists delta:R, delta > 0 /\
forall x':point_set X, dX x x' < delta ->
dY (f x) (f x') < eps.
Lemma metric_space_fun_continuity: forall (X Y:TopologicalSpace)
(f:point_set X->point_set Y) (x:point_set X)
(dX:point_set X -> point_set X -> R)
(dY:point_set Y -> point_set Y -> R),
metrizes X dX -> metrizes Y dY ->
(forall eps:R, eps > 0 -> exists delta:R, delta > 0 /\
forall x':point_set X, dX x x' < delta ->
dY (f x) (f x') < eps) ->
continuous_at f x.