Continuity

The definition of continuity of a function used in most first-year calculus textbooks reads something like this: A function f is continuous at x = a if, and only if, (1) f(a) exists (the value is a finite number), (2)  exists (the limit is a finite number), and (3)  (the limit equals the value). A…

Definitions

Definitions are similar to theorems, but are true in both directions; technically, this means that the statement and its converse are both true (). The double arrow is read “if, and only if.” Both parts are either true or both parts are false. Definitions usually name some thing or some property.  Definitions are not proved.…

Fun with Continuity

Most functions we see in calculus are continuous everywhere or at all but a few points that can be easily identified. But consider the Dirichlet function: Since there is one (actually many) rational numbers between any two irrational number and one (many again) irrational numbers between any two rational numbers, this function is not continuous anywhere! But a…

Continuity

Limits logically come before continuity since the definition of continuity requires using limits. But practically and historically, continuity comes first. The concept of a limit is used to explain the various kinds of discontinuities and asymptotes. Start by studying discontinuities. Types of discontinuities to consider: removable (a gap or hole in the graph), jump, infinite…

Why Limits?

There are four important things before calculus and in beginning calculus for which we need the concept of limit. The first is continuity. Most of the time in pre-calculus mathematics and in the calculus we deal with nice continuous functions or functions that are not continuous at just a few points. Limits give us the…