ABSTRACT
This chapter states formally what it means for a function to be continuous at a point. Functions are continuous nowhere because the limit inferior and superior never match. Functions are continuous at one and only one point because the limit inferior and superior only match at one point. It is also possible to have even stranger functions. Dirichlet’s function fails to be continuous at a countable number of points, not just at a finite number of points. Dirichlet’s function is continuous on the irrationals only. The ideas of continuity and limit are pointwise concepts. The limit of a function exists if and only if its subsequential limits match the limit value. The limit of a function exists with a specific value if and only if right and left hand limits match with value. The chapter discusses the algebra of limits theorem and the algebra of continuity.
