# 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:

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:

### 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

**and**

*f*(a)**, then there is at least one number**

*f*(b)**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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