Everyday usage and use in mathematics
In everyday english interval is an uninterrupted stretch of distance or time. The Oxford English Dictionary [1] provides a number of definitions of interval including "The period of time between two events, actions, etc." and "An open space between two things or two parts of the same thing." For example, the time interval between the signing of the Declaration of Independence and the death of John Adams is 50 years.
In mathematics, an interval is the set of points between two numbers $x$ and $y$. The points $x$ and $y$ might or might not be part of the interval; a square bracket "[" is used if the endpoint lies in the interval, and a round bracket or parenthesis "(" is used if the endpoint is not part of the interval. So the interval $[2, 4.5)$ is the set of points $x$ that are at least $2$ and strictly less than $4.5$. Notice that an interval is a contiguous, uninterrupted piece of the number line. It is possible for $x$ or $y$ to be $\infty$; for example, the interval $(3, \infty)$ is the set of all real numbers that are strictly larger than $3$.
Summary
In everyday english, an interval is an uninterrupted period of time or an uninterrupted stretch of space. In mathematics, an interval is an uninterrupted piece of the number line.
Importance in mathematics
Intervals are frequently used when stating the domain or the range of a function. For example, the domain of the real-valued function $f(x) = \sqrt{4 - x^2}$ is $[-2,2]$ because this function only makes sense when $x$ is at least $-2$ and at most $2$.
Intervals play a special role in mathematics because they are connected subsets of the number line. It is possible to move from any point in an interval to any other point without leaving the interval. This is why results such as the intermediate value theorem hold for intervals.
Exercises
- Consider the sentence "the mice were fed at regular intervals." What do you think "regular intervals" means in this sentence?
- Give an example of a function that is nonnegative on the interval $[-1,1]$ and negative everywhere else.
References
[1] Oxford English Dictionary