In
mathematics, the notion of a quasi-continuous function is similar to, but weaker than, the notion of a
continuous function. All continuous functions are quasi-continuous but the converse is not true in general.
Definition
Let
be a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
. A real-valued function
is quasi-continuous at a point
if for any
and any
open neighborhood
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a p ...
of
there is a non-empty
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
such that
:
Note that in the above definition, it is not necessary that
.
Properties
* If
is continuous then
is quasi-continuous
* If
is continuous and
is quasi-continuous, then
is quasi-continuous.
Example
Consider the function
defined by
whenever
and
whenever
. Clearly f is continuous everywhere except at x=0, thus quasi-continuous everywhere except (at most) at x=0. At x=0, take any open neighborhood U of x. Then there exists an open set
such that
. Clearly this yields
thus f is quasi-continuous.
In contrast, the function
defined by
whenever
is a rational number and
whenever
is an irrational number is nowhere quasi-continuous, since every nonempty open set
contains some
with
.
References
*
* {{cite journal
, jstor=44151947
, author=T. Neubrunn
, title=Quasi-continuity
, journal=Real Analysis Exchange
, volume=14
, number=2
, pages=259–308
, year=1988
Calculus
Theory of continuous functions