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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
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hDefinitioni h26A15i h54C05i h81-00i h82-00i h83-00i h46L05i
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