There are a number of places where the concept of interval is useful. The solution to an inequality in one variable is usually one or more intervals. For example, the solution to 3x + 4 ≤ 10 is the interval (-∞,2].
The interval concept is also useful in calculus. For instance, when a function is said to be continuous on an interval [a,b], it means that the graph of the function is unbroken, no points are missing, and no sudden jumps occur anywhere between x = a and x = b. The concept of interval is also useful in understanding and evaluating integrals. An integral is the area under a curve or graph of a function. An area must be bounded on all sides to be finite, so the area under a curve is taken to be bounded by the function on one side, the x-axis on one side and vertical lines corresponding to the endpoints of an interval on the other two sides.
Bittinger, Marvin L, and Davic Ellenbogen. Intermediate Algebra: Concepts and Applications. 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.
Gowar, Norman. An Invitation to Mathematics. New York: Oxford University Press, 1979.
Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.
J. R. Maddocks