- Topics At a Glance
- Continuity at a Point
- Continuity at a Point via Pictures
- Continuity at a Point via Formulas
- Functions and Combinations of Functions
- Continuity on an Interval
- Continuity on an Interval via Pictures
- Continuity on an Interval via Formulas
- Continuity on Closed and Half-Closed Intervals
- Determining Continuity
- The Informal Version
- The Formal Version
- Properties of Continuous Functions
- Boundedness
- Extreme Value Theorem
- Intermediate Value Theorem
**In the Real World**- Page: I Like Abstract Stuff; Why Should I Care?
- Page: How to Solve a Math Problem
**Appendix: Intervals and Interval Notation**

An **interval** on the real line is the set of all numbers that fall between two specified endpoints.

Let *a* and *b* be real numbers with *a* < *b*. We can have the following types of finite intervals:

- The
**open interval**(*a*,*b*) is the set of all real numbers*x*that fall strictly in between*a*and*b*. That is, all real numbers*x*with*a*<*x*<*b*. The values*a*and*b*are not included in this interval.

- The
**closed interval**[*a*,*b*] is the set of all real numbers*x*that fall (non-strictly) in between*a*and*b*. That is, all real numbers*x*with*a*≤*x*≤*b*. The values*a*and*b*are included in this interval.

- The interval [
*a*,*b*) is the set of all real numbers*x*with*a*≤*x*<*b*.*a*is included in this interval, while*b*is not.

- The interval (
*a*,*b*] is the set of all real numbers*x*with*a*<*x*≤*b*.*b*is included in this interval, while*a*is not.

The values *a* and *b* are called **endpoints** because they're the points at either end of the interval.

To remember which of the two intervals (*a*,*b*) or [*a*,*b*] includes the endpoints *a* and *b*, try thinking of the interval notation like arms. If the arms are like the brackets [*a*,*b*] then they are holding *a* and *b* firmly in there. If arms are like the parentheses (*a*,*b*), then the endpoints *a* and *b* slip out.

The interval (*a*,*b*) is called open, while the interval [*a*,*b*] is called closed. The intervals [*a*,*b*) and (*a*,*b*] are neither open nor closed. We might hear these intervals called "half-closed," "semi-closed," "half-open", or "semi-open".

We can also have infinite intervals. Since ∞ isn't a number, we'll always have parentheses around ∞ or ∞, not the closed brackets. Here are the types of infinite intervals we can have, assuming that *a* is some finite number:

- (-∞, ∞), which is the whole real line. In other words, this is the set of all real numbers.

- (-∞,
*a*) is the set of real numbers*x*with*x*<*a*. This interval does not include*a*.

- (-∞,
*a*] is the set of real numbers*x*with*x*≤*a*. This interval does include*a*.

- (
*a*,∞) is the set of real numbers*x*with*a*<*x*. This interval does not include*a*.

- [
*a*,∞) is the set of real numbers*x*with*a*≤*x*. This interval does include*a*.

The infinite intervals (-∞, ∞), (-∞, *a*], and [*a*,∞) are closed intervals. The infinite intervals (-∞,*a*) and (*a*,∞) are open.

The infinite intervals (-∞, ∞), (-∞, *a*], and [*a*,∞) are closed intervals. The infinite intervals (-∞,*a*) and (*a*,∞) are open.