Almost continuity

A generic term used to describe any condition on a function $f$ such that all continuous functions satisfy it; one can also use it if the original $f$ is not necessarily continuous.

The original condition of continuity by A. Cauchy [a2] was cleared by K. Weierstrass (late 1850s) from the vagueness of its formulation as well as its dependence upon motion (cf. also Continuous function). One of the first conditions of "almost continuity" was the Lipschitz condition, introduced in 1864; Riemann-integrable functions were studied in 1867 (cf. also Riemann integral), while in 1870 H. Hankel introduced pointwise discontinuous functions (cf. Discontinuity point; Discontinuous function).

Nowadays (2000), the term "almost continuity" is used for various conditions weakening the (topological) condition of continuity that the inverse image of any open set is open. For example, V. Volterra noticed that all real-valued separately continuous functions from the plane have a certain almost continuity property, which was later termed quasi-continuity, where it is required that the inverse image of every open set is semi-open, i.e., is contained in the closure of its interior; quasi-continuity has been successfully used in recent proofs of "deep" results in topological algebra (cf. also Separate and joint continuity), in particular in the proof that all Čech-complete semi-topological groups are topological (A. Bouziad, [a1]). Another frequently used type of almost continuity is the notion of near continuity, introduced by B.J. Pettis; it is used in place of linearity in topological versions of the closed-graph theorem, where the spaces under consideration are not necessarily assumed to be linear [a4].

The papers [a5] and [a3] serve as good guides in this rapidly growing field.

References

[a1]	A. Bouziad, "Every Čech-analytic Baire semitopological group is a topological group" Proc. Amer. Math. Soc. , 124 (1996) pp. 953–959
[a2]	A.L. Cauchy, "Cours d'analyse d'École Royale Polytechnique, 1821" , Oeuvres Complétes d'Augustin Cauchy, II Ser. , III , Gauthier-Villars (1897)
[a3]	D. Gauld, S. Greenwood, I. Reilly, "On variations of continuity" Invited Contribution, Topology Atlas (2000) (http://at.yorku.ca/t/a/i/c/32.htm)
[a4]	Z. Piotrowski, A. Szymański, "Closed graph theorem: Topological approach" Rend. Circ. Mat. Palermo , 37 (1988) pp. 88–99
[a5]	M. Przemski, "On forms of continuity and cliquishness" Rend. Circ. Mat. Palermo , 42 (1993) pp. 417–452