# Boundedness

## Key Questions

Boundedness is about having finite limits. In the context of values of functions, we say that a function has an upper bound if the value does not exceed a certain upper limit. More...

#### Explanation:

Other terms used are "bounded above" or "bounded below".

For example, the function $f \left(x\right) = \frac{1}{1 + {x}^{2}}$ is bounded above by $1$ and below by $0$ in that:

$0 < f \left(x\right) \le 1$ for all $x \in \mathbb{R}$

graph{1/(1+x^2) [-5, 5, -2.5, 2.5]}

The function $\exp : x \to {e}^{x}$ is bounded below by $0$ (or you can say has $0$ as a lower bound), but is not bounded above.

$0 < {e}^{x} < \infty$ for all $x \in \mathbb{R}$

graph{e^x [-5.194, 4.806, -0.74, 4.26]}

A continuous function defined on a closed interval has an upper (and lower) bound.

#### Explanation:

Probably the simplest boundedness theorem states that a continuous function defined on a closed interval has an upper (and lower) bound.

Suppose $f \left(x\right)$ is defined and continuous on a closed interval $\left[a , b\right]$, but has no upper bound.

Then:

$\forall n \in \mathbb{N} , \exists {x}_{n} \in \left[a , b\right] : f \left({x}_{n}\right) > n$

Since the sequence of ${x}_{n}$'s lies in a bounded interval, it is dense at some point in the closure of the interval. Since the interval is closed, that must be at some point $c$ actually in the interval $\left[a , b\right]$.

Since the sequence of ${x}_{n}$'s is dense at $c$, there is some monotonically increasing sequence ${n}_{k} \in \mathbb{N}$ such that ${x}_{{n}_{k}} \to c$ as $k \to \infty$.

Now $f \left(x\right)$ is continuous at $c$, so:

${\lim}_{x \to c} f \left(x\right) = f \left(c\right)$

which is bounded.

But:

${\lim}_{k \to \infty} {x}_{{n}_{k}} = c \text{ }$ and $\text{ } {\lim}_{k \to \infty} f \left({x}_{{n}_{k}}\right) = \infty$

is unbounded.

So there is no such $f \left(x\right)$ lacking upper (or lower) bound.

• If the function is unbounded, the graph would progress to infinity, in some direction(s).

• Not all functions are bounded.

The simplest counter example would be
the identity function $f \left(x\right) = x$
which is defined for all values of $x$ and can generate any value for $f \left(x\right)$

A slightly less trivial counter example would be
the cubing function $f \left(x\right) = {x}^{3}$