Assume that you have developed a good set of tools allowing you to decide which real functions of one real variable, f:R -->R, are continuous. (Q): How can such a tool-box be utilized to decide on the continuity of the functions g:Rn -->R of n real variables? This is one of the questions which must be faced by any student taking multivariable calculus. Of course, such a student is following the footsteps of many generations of mathematicians, which were, and still are, struggling with the same general question. The aim of this article is to present the history and the current research related to this subject in a real analysis perspective, rather than in a more general, topological perspective.
In addition to surveying the results published so far, this exposition includes also several original results (Theorems 1, 11 and 12), as well as some new simplified versions of the (sketches of the) proofs of older results. We also recall several intriguing open problems.
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Last modified April 21, 2016.