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global versus local continuity∗
CWoo†
2013-03-22 3:21:27
In this entry, we establish a very basic fact about continuity:
Proposition 1. A function f : X → Y between two topological spaces is continuous iff it is continuous at every point x ∈ X.
Proof. Suppose first that f is continuous, and x ∈ X. Let f (x) ∈ V be an open
set in Y . We want to find an open set x ∈ U in X such that f (U ) ⊆ V . Well,
let U = f −1 (V ). So U is open since f is continuous, and x ∈ U . Furthermore,
f (U ) = f (f −1 (V )) = V .
On the other hand, if f is not continuous at x ∈ X. Then there is an open
set f (x) ∈ V in Y such that no open sets x ∈ U in X have the property
f (U ) ⊆ V.
(1)
Let W = f −1 (V ). If W is open, then W has the property (1) above, a contradiction. Since W is not open, f is not continuous.
∗ hGlobalVersusLocalContinuityi created: h2013-03-2i by: hCWooi version: h42056i
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