Continuity

Continuity is an important but simple concept. Basically, a continuous function is one that you could draw without having to lift your pencil. A function can be continuous throughout its entire domain, or only on a particular interval. The following are examples of functions that are not continuous:

Graph of f(x)=sqrt(x-1)/(x-1)

Graph of g(x) showing the removable discontinuity at x=1
The second graph shows a function with a removable discontinuity. This type of function could be made continuous by “filling it in” at one point.

In general a function, f(x), is continuous at some point, x = a, as long as 3 conditions are satisfied:

  1. f(a) is defined.
  2. The limit of f(X) as x approaches a exists and is finite.

  3. The limit of f(x) as x approaches a equals f(a).

Intermediate Value Theorem

This theorem is a good one to know because it shows up in many proofs.

Suppose that a function, f, is continuous on an interval that includes the points a and b with a < b and that L is a number between f(a) and f(b), then there is at least one number x with a ≤ x ≤ b such that f(x) = L.


Sources

Shenk, A. (1988). Calculus and Analytic Geometry [4th Ed.]. Glenview, IL: Scott Foresman and Company.

Swokowski, E., Olinick, M., Pence, D. & Cole, J. (1994). Calculus of a Single Variable [6th Ed.]. Boston, MA: PWS.


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