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continuous∗ djao† 2013-03-21 12:36:10 Let X and Y be topological spaces. A function f : X → Y is continuous if, for every open set U ⊂ Y , the inverse image f −1 (U ) is an open subset of X. In the case where X and Y are metric spaces (e.g. Euclidean space, or the space of real numbers), a function f : X → Y is continuous if and only if for every x ∈ X and every real number > 0, there exists a real number δ > 0 such that whenever a point z ∈ X has distance less than δ to x, the point f (z) ∈ Y has distance less than to f (x). Continuity at a point A related notion is that of local continuity, or continuity at a point (as opposed to the whole space X at once). When X and Y are topological spaces, we say f is continuous at a point x ∈ X if, for every open subset V ⊂ Y containing f (x), there is an open subset U ⊂ X containing x whose image f (U ) is contained in V . Of course, the function f : X → Y is continuous in the first sense if and only if f is continuous at every point x ∈ X in the second sense (for students who haven’t seen this before, proving it is a worthwhile exercise). In the common case where X and Y are metric spaces (e.g., Euclidean spaces), a function f is continuous at x ∈ X if and only if for every real number > 0, there exists a real number δ > 0 satisfying the property that dY (f (x), f (z)) < for all z ∈ X with dX (x, z) < δ. Alternatively, the function f is continuous at a ∈ X if and only if the limit of f (x) as x → a satisfies the equation lim f (x) = f (a). x→a ∗ hContinuousi created: h2013-03-21i by: hdjaoi version: h30439i Privacy setting: h1i hDefinitioni h26A15i h54C05i h81-00i h82-00i h83-00i h46L05i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1