Bounded Function

Q: Can a function be bounded above but not bounded below?

Yes. A function that is bounded above has an upper bound $M$ with $f(x) \leq M$ for all $x$, but it might decrease without limit. For instance, $f(x) = -x^2$ is bounded above by $0$ but has no lower bound. To be a bounded function, the function must be bounded both above and below.

Bounded Function

A function with a range that is a bounded set. The range must have both an upper bound and a lower bound.

Graph of a bounded function on x-y axes, showing an S-shaped curve with a labeled "range" indicated by a vertical arrow on the...

Key Formula

m \leq f(x) \leq M \quad \text{for all } x \text{ in the domain}

Where:

$f(x)$ = The function being evaluated
$x$ = Any input value from the function's domain
$m$ = A real number that serves as a lower bound on the function's output
$M$ = A real number that serves as an upper bound on the function's output

Worked Example

Problem: Show that the function f(x) = sin(x) is bounded.

Step 1: Recall the range of the sine function. For every real number

x

, the output of

\sin(x)

lies between

-1

and

1

-1 \leq \sin(x) \leq 1

Step 2: Identify the lower bound

m

and upper bound

M

. Here

m = -1

and

M = 1

m = -1, \quad M = 1

Step 3: Verify the definition: for every

x

in the domain (all real numbers),

f(x)

satisfies the inequality.

-1 \leq f(x) \leq 1 \quad \text{for all } x \in \mathbb{R}

Step 4: Since both a lower bound and an upper bound exist, the function is bounded.

Answer: f(x) = sin(x) is a bounded function with lower bound −1 and upper bound 1.

Another Example

This example shows that a function which is unbounded on all of ℝ (since x² grows without limit) can become bounded when its domain is restricted to a closed interval.

Problem: Determine whether the function g(x) = x² on the domain [−3, 3] is bounded, and if so, find bounds.

Step 1: Identify the domain. Here the domain is restricted to

[-3, 3]

, not all real numbers.

x \in [-3, 3]

Step 2: Find the minimum output. Since

x^2 \geq 0

for all

x

, and

g(0) = 0

, the minimum output is

0

g(0) = 0^2 = 0

Step 3: Find the maximum output. The largest values of

x^2

[-3,3]

occur at the endpoints

x = -3

and

x = 3

g(3) = 3^2 = 9, \quad g(-3) = (-3)^2 = 9

Step 4: Check the definition with

m = 0

and

M = 9

0 \leq g(x) \leq 9 \quad \text{for all } x \in [-3, 3]

Answer: g(x) = x² on [−3, 3] is bounded, with lower bound 0 and upper bound 9.

Frequently Asked Questions

What is the difference between a bounded function and an unbounded function?

A bounded function has outputs that stay within a finite range — there exist real numbers

m

and

M

with

m \leq f(x) \leq M

for every input

x

. An unbounded function has no such pair of bounds; its outputs can grow arbitrarily large in the positive direction, the negative direction, or both. For example,

f(x) = \sin(x)

is bounded, while

f(x) = x^3

is unbounded.

Can a function be bounded above but not bounded below?

Yes. A function that is bounded above has an upper bound

M

with

f(x) \leq M

for all

x

, but it might decrease without limit. For instance,

f(x) = -x^2

is bounded above by

0

but has no lower bound. To be a bounded function, the function must be bounded both above and below.

Is every function on a closed interval bounded?

Not necessarily in general, but every continuous function on a closed interval

[a, b]

is guaranteed to be bounded. This is a famous result called the Extreme Value Theorem. However, a discontinuous function on a closed interval can still be unbounded — for example,

f(x) = 1/x

[-1, 1]

(excluding

0

, or with any assigned value at

0

) is unbounded near

x = 0

Bounded Function vs. Unbounded Function

	Bounded Function	Unbounded Function
Definition	There exist real numbers m and M such that m ≤ f(x) ≤ M for all x in the domain	No such pair m, M exists; the outputs can grow without limit
Key condition	\|f(x)\| ≤ K for some constant K and all x	For any K > 0, there exists some x with \|f(x)\| > K
Examples	sin(x), cos(x), 1/(1 + x²)	x², eˣ, ln(x), tan(x)
Graphical appearance	Graph stays within a horizontal strip	Graph eventually escapes any horizontal strip

Why It Matters

Bounded functions appear throughout calculus and analysis. The Extreme Value Theorem guarantees that continuous functions on closed intervals are bounded and attain their maximum and minimum — a fact used constantly in optimization problems. In sequences and series, knowing whether terms come from a bounded function is essential for applying convergence tests like the squeeze theorem.

Common Mistakes

Mistake: Claiming a function is bounded just because it has an upper bound, while ignoring the lower bound (or vice versa).

Correction: A function must have both an upper bound and a lower bound to be called bounded. Having only one is called 'bounded above' or 'bounded below,' not simply 'bounded.'

Mistake: Assuming f(x) = x² is bounded because its outputs are always non-negative (bounded below by 0).

Correction: While x² is bounded below, it has no upper bound on the domain of all real numbers since x² → ∞ as x → ±∞. A function needs both bounds to qualify as bounded.

Related Terms

Function — The general concept being classified as bounded
Range — The set of output values that must be bounded
Bounded Set of Numbers — A bounded function's range forms a bounded set
Upper Bound of a Set — The value M that caps the function's outputs
Lower Bound of a Set — The value m that floors the function's outputs
Domain — Restricting the domain can make a function bounded
Continuous Function — Continuous functions on closed intervals are always bounded