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In topology, a continuous function is generally defined as one
for which preimages of open sets are
open. Continuous functions are fundamental in describing the relationships between
topological spaces, and allow simple generalizations of many
results from real analysis to be proven. Because this definition only
"uses" open sets, this makes continuity of a function a topological property, depending only on the topologies
of its domain and range spaces.
Formulations of Continuity
Several equivalent formulations of continuity can be made, and each is useful in different situations. Similar to the open set
formulation is the closed set formulation, which says that preimages of closed sets are closed.
Definition based on preimages are often difficult to use directly. Instead, suppose we have a function f from
X to Y, where X,Y are topological spaces. We say f is continuous
at x for some if for any
neighborhood V of
f(x), there is a neighborhood U of x such that . Although this definition appears complex, the intuition
is that no matter how "small" V becomes, we can find a small U containing x that will map inside
it. If f is continuous at every ,
then we simply say f is continuous.
In a metric space, it is equivalent to consider only open balls centered at x and f(x) instead of all
neighborhoods. This leads to the standard delta-epsilon definition of a continuous function from real analysis, which says roughly that a function is continuous if all points close to
x map to points close to f(x). This only really makes sense in a metric sense, however, which
has a notion of closeness.
Useful properties of continuous maps
Some facts about continuous maps between topological spaces:
- If f : X → Y and g : Y → Z are continuous, then
so is the composition g o f : X → Z.
- If f : X → Y is continuous and
- If f : X → Y and X is a metric space, then we also have:
- If a sequence (xn) converges to a limit x, then
the sequence (f(xn)) obtained by applying f to each element converges to
f(x). We say continuous functions take limits to limits. When using nets instead of sequences, this holds for a general topological space X.
Other notes
If a set is given the discrete topology, all functions with
that space as a domain are continuous. If the domain set is given the trivial topology, a topology with only two open sets, and the range set is T1, then only constant functions are continuous.
Symmetric to the concept of a continuous map is an open map, for which
images of open sets are open. In fact, if an open map f has an inverse, that inverse is continuous, and if a
continuous map g has an inverse, that inverse is open.
If a function is a bijection, then it has an inverse function. The inverse of a continuous bijection need not be continuous, but if it is, this
special function is called a homeomorphism.
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