 # Bounded Function & Unbounded: Definition, Examples

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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

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 Real numbers (ℝ) include the rational (ℚ), which include the integers (Z), which include the natural numbers (N).

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 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 xM we have xM.  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.).

## Bounded Function: References

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