| The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages. |
|
Home  Continuous Functions Continuity Continuity on an Interval |
|||||
 Search the VIAS Library | Index
|
  |
||||
Continuity on an Interval
Functions which are "continuous on an interval" will play an important role in this chapter. Intervals were discussed in Section 1.1. Recall that closed intervals have the form [a,b] open intervals have one of the forms (a, b), (a, ∞), (-∞, b), (-∞, ∞), and half-open intervals have one of the forms [a, b) (a, b], [a, ∞), (-∞, b]. In these intervals, a is called the lower endpoint and b, the upper endpoint. The symbol -∞ indicates that there is no lower endpoint, while ∞ indicates that there is no upper endpoint.
DEFINITION We say that f is continuous on an open interval I if f is continuous at every point c in I. If in addition f has a derivative at every point of I, we say that f is differentiable on I. To define what is meant by a function continuous on a closed interval, we introduce the notions of continuous from the right and continuous from the left, using one-sided limits. DEFINITION f is continuous from the right at c if limx→c+ f(x) = f(c). f is continuous from the left at c if limx→c- f(x) = f(c). DEFINITION f is said to be continuous on the closed interval [a, b] if f is continuous at each point c where a < c < b, continuous from the right at a, and continuous from the left at b. Figure 3.4.11 shows a function f continuous on [a, b].
Figure 3.4.11: f is continuous on the interval [a, b]
|
|||||
| Home Continuous Functions Continuity Continuity on an Interval |
|
||||
Last Update: 2006-11-24