# How to explain what it means to say a function is "defined" on an interval?

I am having difficulty in explaining the terminology "defined" to the students I am assisting. Here is the sentence: If a real-valued function $f$ is defined and continuous on the closed interval $[a,b]$ in the real line, then $f$ is bounded on $[a,b]$. Can I have some thoughts on how to explain the word "defined" used in the sentence? Thank you.

• $f$ is defined on the interval $[a, b],$ means that we know $f$ (either its value, its expression, or how to compute it) for every $x \in [a, b].$ Outside this interval, we know nothing about $f.$
– user2468
Jul 5, 2012 at 17:26
• ...or, put another way, the function has no discontinuities within the closed interval being considered. Jul 5, 2012 at 17:27
• "$f$ is defined on the closed interval $[a,b]$" means the 'domain' of $f$ is $[a,b]$ (Note that 'domain','codomain' are necessary ingredients to describe any function). Jul 5, 2012 at 17:28
• Great. Thanks Matt, J.D., J.M. and Pritam. Now I got an idea on how to explain it. Jul 5, 2012 at 17:31

What age students? If it's just a precalculus or calculus course, I would just give examples of a nice looking formula that "isn't defined" on all of an interval, e.g. $\log(x)$ on [-.5, 2] or $1/x$ on [-1, 1].
If it's an analysis course, I would interpret the word defined in this sentence as saying, "there's some function $f$, taking values in $\mathbb{R}$, whose domain is a subset of $\mathbb{R}$, and whatever the domain is, definitely it includes the closed interval $[a, b]$."
A function is a domain $A$ and a codomain $B$ and a subset $f \subset A\times B$ with the property that if $(x,y)$ and $(x,y')$ are both in $f$, then $y=y'$ and that for every $x \in A$ there is some $y \in B$ such that $(x,y) \in f$. We write $f : A \to B$. If $(x,y) \in f$, we write $f(x) = y$. We may say, for any set $S \subset A$ that $f$ is defined on $S$.