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## 1. What is a Bounded Function?

**Bounded functions** have some kind of boundaries or constraints placed upon them. Most things in real life have natural bounds: cars are somewhere between 6 and 12 feet long, people take between 2 hours and 20 hours to complete a marathon, cats range in length from a few inches to a few feet. When you place those kinds of bounds on a function, it becomes a *bounded function*.

In order for a function to be classified as “bounded”, its range must have both a **lower bound** (e.g. 7 inches) **and **an **upper bound** (e.g. 12 feet).

Any function that isn’t bounded is **unbounded**. A function can be bounded at one end, and unbounded at another.

## Upper Bound for a Bounded Function

If a function only has a range with an upper bound (i.e. the function has a number that fixes how high the range can get), then the function is called *bounded from above*. Usually, the lower limit for the range is listed as -∞.

More formally, an upper bound is defined as follows:

A set

A∈ ℝ of real numbers is bounded from above if there exists a real number M ∈ R, called an upper bound of A, such that x ≤ M for every x ∈ A (Hunter, n.d.).

Basically, the above definition is saying there’s a real number, M, that we’ll call an **upper bound**. Every element in the set is lower than this value M. Don’t get confused by the fact that the formal definition uses an “x” to denote the elements in the set; It doesn’t mean x-values (as in, the domain). The definition of bounded only applies to the range of values a function can output, **not **how high the x-values can get.

The exact definition is slightly different, depending on where you’re using the term.

## 1. Upper Bounded Function or Set

The **upper bound of a function (U)** is that function’s largest number. More formally, you would say that a function *f* has a *U* if f(x) ≤ U for all *x* in the function’s domain.

If you’re working with an **interval **(i.e. a small piece of the function), then U on the interval is the largest number in the interval. In notation, that’s:

f(x) ≤ U for all x on [a, b].

In the same way, the **upper bound of a set (U)**is the largest number in the set. In other words, it’s a number that’s greater than or equal to all of the elements in the set. For example, 132 is U for the set { 3, 7, 39, 75, 132 }.

## Integration

The upper bound of an integral is the where you stop integrating. It’s above the integral symbol:

See: Integral Bounds.

## 3. Use in Estimation

In estimation, an “upper bound” is the **smallest value that rounds up to the next value**.

For example, let’s say you had an object that was 7 cm long, rounded to the nearest cm. The upper bound is 7.5 cm, because 7.5 cm is the smallest length that would round up to the next increment—8 cm. Similarly, a lower bound is the smallest value that rounds up to 7cm— 6.5 cm.

You’re stating that the 7 cm object is actually anywhere between 6.5 cm (the lower bound) and 7.5 cm (the upper bound).

## Least Upper Bound of a Bounded Function

**Least upper bound (LUB) **refers to a number that serves as the lowest possible ceiling for a set of numbers.

**If a set of numbers has a greatest number,** then that number is also the least upper bound (supremum). For example, let’s say you had a set defined by the closed interval [0,2]. The number 2 is included in the set, and is therefore the least upper bound.

Where things get a little interesting is when **a set of numbers doesn’t have an upper bound**. In that case, the supremum is the number that “*wants to be* the greatest element” (Howland, 2010). Take the open interval {0,2}. Although the set is bounded by the number 0 and 2, they aren’t actually in the set. However, 2 wants to be the greatest element, and so it’s the least upper bound.

## When The Least Upper Bound Doesn’t Exist

Any set of real numbers ordered with < has a least upper bound. Some sets don’t have a supremum. For example (Holmes, n.d.):

**Rational numbers**ordered by <. Let’s say you had a set of rational numbers where all the elements are less than √2. You can find an upper bound (e.g. the number 2), but the only candidate for the least upper bound is √2, and that number isn’t a rational number (it’s a real number). And a real number can’t be the supremum for a set of rational numbers.^{*}- If a set has
**no upper bound**, then that set has no supremum. For example, the set of all real numbers is unbounded. - The
**empty set**doesn’t have a least upper bound. That’s because*every number*is a potential upper bound for the empty set.

^{*}The rational numbers pose all kinds of problems like this that render them “…unfit to be the basis of calculus” (Bloch, p.64).

## More Formal Definition

In the case of the open interval {0,2}, the number is is the smallest number that is larger than every member in the set. In other words, 2 isn’t actually in the set itself, but it’s the smallest number outside of the set that’s larger than 1.999….

In more formal terms:

If** M** is a set of numbers and *M *is a number, we can say that *M* is the least upper bound or supremum of **M** if the following two statements are true:

*M*is an upper bound of**M**, and- no element of
**M**which is less than*M*can be an upper bound for**M**.

Assume that *M* is the least upper bound for **M**. What this means is that for every number *x* ∈ **M **we have *x* ≤ *M*. For any set of numbers that has an upper bound, the set is *bounded from above*.

## Lower Bound

If a function has a range with a lower bound, it’s called *bounded from below*. Usually, the lower limit for the range is listed as +∞. The formal definition is almost the same as that for the upper bound, except with a different inequality.

A set

A∈ ℝ of real numbers is bounded from below if there exists a real number M ∈ R, called a lower bound of A, such that x ≥ M for every x ∈ A (Hunter, n.d.).

A **bounded sequence **is a special case of a bounded function; one where the absolute value of every term is less than or equal to a particular real, positive number. You can think of it as there being a well defined boundary line such that no term in the sequence can be found on the outskirts of that line.

More formally, a sequence X is bounded if there is a real number, M greater than 0, such |x_{n}| ≤ M for all *n* ∈ N.

The blue dots on the image below show an infinite sequence. As you can see, the sequence does not converge, for the red boundary lines never converge. However, it is bounded.

## Examples of Bounded Sequences

One example of a sequence that is bounded is the one defined by”

The right hand side of this equation tells us that n is indexed between 1 and infinity. This makes the sequence into a sequence of fractions, with the numerators always being one and the denominators always being numbers that are greater than one. A basic algebraic identity tells us that x^{-k} = 1 / x^{k}. So each term in the sequence is a fractional part of one, and we can say that for every term in our sequence, |x_{n}| ≤ 1.

Remember now our definition of a bounded sequence:* a sequence X is bounded if there is a real number, M greater than 0, such |x _{n}| ≤ M for all n ∈ N*. Let M = 1, and then M is be a real number greater than zero such that |x

_{n}| ≤ M for all n between 1 and infinity. So our sequence is bounded.

## Bounded Sequences and Convergence

Every absolutely convergent sequence is bounded, so if we know that a sequence is convergent, we know immediately that it is bounded. Note that this doesn’t tell us anything about whether a bounded sequence is convergent: it may or may not be. As an example, the sequence drawn above is not convergent though it is bounded.

## Bounded Above and Below

If we say a sequence is bounded, it is bounded above and below. Some sequences, however, are only bounded from one side.

If all of the terms of a sequence are greater than or equal to a number K the sequence is bounded below, and K is called the lower bound. The greatest possible K is the **infimum**.

If all the terms of a sequence are less than or equal to a number K’ the sequence is said to be bounded above, and K’ is the upper bound. The least possible K is the **supremum**.

## Bounded Function and Bounded Variation

A bounded function of **bounded variation** (also called a *BV function*) “wiggles” or oscillates between bounds, much in the same way that a sine function wiggles between bounds of 1 and -1; The vertical (up and down movement) of these functions is restricted over an interval. In other words, the variation isn’t infinite: we can calculate a value for it.

These functions can be described as integrable functions with a derivative (in the sense of distributions) that is a signed measure with finite total variation [1]. The concept was originally developed in the context of Fourier series [2], when mathematicians were trying to prove the series convergence.

## Examples of Functions of Bounded Variation

All monotonic functions and absolutely continuous functions are of bounded variation; Real‐valued functions with a variation on a compact interval can be expressed as the difference between two monotone (non-decreasing) functions [3], called a *Jordan decomposition.* Interestingly, these functions do not have to be continuous functions and can have a finite number of discontinuities (although they do have to be Riemann integrable). They can also be approximated by finite step functions, or decomposed to part continuous and part jump.

Normalized functions can be described as having bounded variation when on the interval [0,1] with h(0) = 0 and h(c) = h(c + 0) for 0 < c < 1.

More formally, a real-valued function α of bounded variation on the closed interval [a, b] has a constant M > 0 such that [4]:

It’s not always necessary to specific the interval, especially when the interval in question is obvious [5].

## References (Bounded Variation)

[1] Ziemer W.P. (1989) Functions of Bounded Variation. In: Weakly Differentiable Functions. Graduate Texts in Mathematics, vol 120. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1015-3_5

[2] Monteiro, G. et al. Series in Real Analysis. Volume 15-Kurzweil–Stieltjes Integral: Theory and Applications. World Scientific.

[3] Bridges, D. (2016). A Constructive Look at Functions of BV. Bulletin of the London Mathematical Society. Volume 32, Issue 3 p. 316-324

[4] Bridges, D. (2016). Functions of Bounded Variation. retrieved April 8, 2021 from:

http://www.math.ubc.ca/~feldman/m321/variation.pdf

[5] Functions of BV. Retrieved April 8, 2021 from: https://www.diva-portal.org/smash/get/diva2:5850/FULLTEXT01.pdf

## Other Bounded Function References

Bloch, E. (2011). The Real Numbers and Real Analysis. Springer Science and Business Media.

Gallup, Nathaniel. Mat25 Lecture 9 Notes: Boundedness of Sequences. Retrieved from https://www.math.ucdavis.edu/~npgallup/m17_mat25/lecture_notes/lecture_9/m17_mat25_lecture_9_notes.pdf on January 25, 2018.

Holmes (n.d.). Class Notes. Retrieved January 16, 2018 from: https://math.boisestate.edu/~holmes/math314/M314F09lubnotes.pdf

Howland, J. (2010). Basic Real Analysis. Jones & Bartlett Learning.

Hunter, J. Supremum and Infinim. Retrieved December 8, 2018 from: https://www.math.ucdavis.edu/~hunter/m125b/ch2.pdf

Larson & Edwards. Calculus.

Laval, P. Bounded Functions. Retrieved December 8, 2018 from: http://ksuweb.kennesaw.edu/~plaval/math4381/real_bdfunctions.pdf

King, M. & Mody, N. (2010). Numerical and Statistical Methods for Bioengineering: Applications in MATLAB. Cambridge University Press.

Math Learning Center: Sequences. Retrieved from https://www3.ul.ie/cemtl/pdf%20files/cm2/BoundedSequence.pdf on January 26, 2018

Mac Lane et al. (1991). Algebra. Providence, RI: American Mathematical Society. p. 145. ISBN 0-8218-1646-2.

Woodroofe, R. Math 131. Retrieved October 18, 2018 from: https://www.math.wustl.edu/~russw/s09.math131/Upper%20bounds.pdf

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