Continuity at a point Continuous function

the translation in language of neighborhoods of (ε, δ)-definition of continuity leads following definition of continuity @ point:

this definition equivalent same statement neighborhoods restricted open neighborhoods , can restated in several ways using preimages rather images.

also, every set contains neighborhood neighborhood, ,




f

−
1


(
v
)


{\displaystyle f^{-1}(v)}

largest subset u of x such f(u) ⊆ v, definition may simplified into:

as open set set neighborhood of points, function



f
:
x
→
y


{\displaystyle f:x\rightarrow y}

continuous @ every point of x if , if continuous function.

if x , y metric spaces, equivalent consider neighborhood system of open balls centered @ x , f(x) instead of neighborhoods. gives above δ-ε definition of continuity in context of metric spaces. in general topological spaces, there no notion of nearness or distance. if target space hausdorff space, still true f continuous @ if , if limit of f x approaches f(a). @ isolated point, every function continuous.