closed interval

(noun)

A set of real numbers that includes both of its endpoints.

Related Terms

Examples of closed interval in the following topics:

  • Interval Notation

    • A closed interval includes its endpoints and is denoted with square brackets rather than parentheses.
    • The image below illustrates open and closed intervals on a number line.
    • Bounded intervals are also commonly known as finite intervals.
    • For example, the interval $(1,10)$ is considered bounded; the interval $(- \infty, + \infty)$ is considered unbounded.
    • Representations of open and closed intervals on the real number line.

  • Piecewise Functions

    • A piecewise function is defined by multiple subfunctions that are each applied to separate intervals of the input
    • Notice the open and closed circles in the graph.  
    • An open circle at the end of an interval means that the end point is not included in the interval, i.e. strictly less than or strictly greater than.
    • A closed circle means the end point is included (equal to).
    • Since there is an closed AND open dot at $x=1$ the function is piecewise continuous there.  

  • Common Bases of Logarithms

    • Logarithms are related to musical tones and intervals.
    • The interval between A and B-flat is a semitone, as is the one between B-flat and B (frequency 493 Hz).
    • Therefore, logarithms can be used to describe the intervals: an interval is measured in semitones by taking the base-21/12 logarithm of the frequency ratio.
    • Natural logarithms are closely linked to counting prime numbers (2, 3, 5, 7, 11, ...), an important topic in number theory.
    • The prime number theorem states that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N).

  • Increasing, Decreasing, and Constant Functions

    • We say that a function is increasing on an interval if the function values increase as the input values increase within that interval.
    • Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval.
    • There are no intervals where this curve is constant.
    • The interval notation is written as: $(−∞, −2)∪(2, ∞)$.  
    • The function is decreasing on on the interval: $ (−2, 2)$.  

  • Relative Minima and Maxima

    • A function $f$ has a relative (local) maximum at  $x=b$ if there exists an interval  $(a,c)$ with $ainterval $(a,c)$, $f(x)≤f(b)$.
    • Likewise, $f$ has a relative (local) minimum at $x=b$ if there exists an interval $(a,c)$ with $ainterval $(a,c)$, $f(x)≥f(b)$.
    • The graph attains a local maximum at $(1,2)$ because it is the highest point in an open interval around $x=1$.  
    • The graph attains a local minimum at $(-1,-2)$ because it is the lowest point in an open interval around $x=-1$. 

  • Tangent as a Function

    • As with the sine and cosine functions, tangent is a periodic function; its values repeat at regular intervals.
    • The period of the tangent function is $\pi$ because the graph repeats itself on intervals of $k\pi$ where $k$ is a constant.
    • If we look at any larger interval, we will see that the characteristics of the graph repeat.

  • Inverse Trigonometric Functions

    • For angles in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$, if $\sin y = x$, then $\sin^{−1} x=y$.
    • For angles in the interval $[0, \pi]$, if $\cos y = x$, then $\cos^{-1} x = y$.
    • For angles in the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$, if $\tan y = x$, then $\tan^{-1}x = y$.

  • Sine and Cosine as Functions

    • By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval [-1, 1].
    • A periodic function is a function with a repeated set of values at regular intervals.

  • The Intermediate Value Theorem

    • Stated in the language of algebra supported by : If f is a real-valued continuous function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such that f(c) = u.

  • Rational Inequalities

    • Thus we can conclude that for $x$ values on the open interval from $-\infty$ to $-3$, the rational expression is negative.