A function is called continuous if its values do not “jump” with variation of its argument , unless itself “jumps”. Roughly speaking, if , then . (This can be made into a precise definition in nonstandard analysis if care is taken about the domains of these variables.)
In order to make this precise (in standard analysis) one needs some concept of neighbourhoods of elements of and .
For instance if and carry structure of metric spaces, then one may say that is continuous if for every point and for every small open ball around its image in , there exists a sufficiently small open ball around which is still mapped by into that target open ball. This definition turns out to have more elegant formulation that needs to mention neither the points of nor the radii of open balls around points: the metric induces a concept of open subsets and is continuous precisely if preimages under of open subsets in are still open subsets in .
This then is the general definition of continuity of a function between topological spaces:
Continuous maps are the homomorphisms between topological spaces. In other words, the collection of topological spaces forms a category, often denoted Top, whose morphisms are the continuous functions.
A metric space is
a set (the “underlying set”);
such that for all :
(epsilontic definition of continuity)
is said to be continuous at a point if for every there exists such that
or equivalently such that
The function is called just continuous if it is continuous at every point .
This definition is equivalent to a more abstract one, which does not explicitly refer to points or radii anymore:
The collection of open subsets in def. 4 constitutes a topology on the set , making it a topological space. This is called the metric topology. Stated more concisely: the open balls in a metric space constitute the basis of a topology for the metric topology.
A function between metric spaces (def. 1) is continuous in the epsilontic sense of def. 3 precisely if it has the property that its pre-images of open subsets of (in the sense of def. 4) are open subsets of .
First assume that is continuous in the epsilontic sense. Then for any open subset and any point in the pre-image, we need to show that there exists a neighbourhood of in . But by assumption there exists an open ball with . Since this is true for all , by definition this means that is open in .
Conversely, assume that takes open subsets to open subsets. Then for every and an open ball around its image, we need to produce an open ball in its pre-image. But by assumption contains a neighbourhood of which by definition means that it contains such an open ball around .
In nonstandard analysis, this is equivalent to
A function between topological spaces is a continuous map (or is said to be continuous) if for every standard point? and every hyperpoint? , if and are adequal? (infinitely close, or in other words if is in the halo of ), then and are adequal (where is the nonstandard extension? of ). Equivalently, is continuous iff is microcontinuous?.
A continuous map between locales is simply a frame homomorphism in the opposite direction. Equivalently (via the adjoint functor theorem), it may be defined as a homomorphism of inflattices whose left adjoint preserves finitary meets (and hence is a frame homomorphism).
Since continuity is defined in terms of preservation of property (namely preserving “openness” under preimages), it is natural to ask what other properties they preserve.
Also, when a property is not always preserved it is useful to label those maps which do preserve it for closer study.
The preimage of a compact set need not be compact; a continuous map for which this is true is known as a proper map.
The image of an closed set need not be closed; a continuous map for which this is true is said to be an closed map. (Technically, a closed map is any function with just this property.)
Although these don’t make sense for arbitrary topological spaces (convergence spaces, locales, etc), they are special kinds of continuous maps in contexts such as metric spaces:
In classical mathematics, these are all equivalent when the domain is itself a closed and bounded interval, and all of them except for uniform continuity are equivalent in general. The same equivalences hold in intuitionistic mathematics, thanks to the fan theorem. But no two of these are equivalent in Russian constructivism.
In fact, assuming that is defined as the set of located Dedekind cuts, there is the following negative result by Frank Waaldijk (Waaldijk2003): Without the fan theorem, there is no notion of continuity for set-theoretic functions in constructive mathematics, spelled “kontinuity” in the following, such that all of the following desiderata are met:
The key problem is that a uniformly continuous, positive-valued function defined on might fail to be bounded below by a positive number, since the interval might fail to be compact, yet its reciprocal (if also uniformly continuous) must be bounded above.