What is the precise definition of 'between'?

I’m wondering what the precise definition of ‘between’ is in a mathematical context. For example, many statements of the intermediate value theorem state that a value $k$ is ‘between’ $f(a)$ and $f(b)$. Would this mean that $f(a)≤k≤f(b)$ or $f(a)<k<f(b)$? Thanks for the clarification.

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There is no standard meaning of “between”; either of the two definitions you suggest is possible. Sometimes “strictly between” is used to mean the version with strict inequalities. The word “between” should be avoided when making precise statements unless you explicitly clarify the meaning (or unless only one meaning could possibly make sense in context).

Just “between $f(a)$ and $f(b)$”, for example, means in majority of cases a non-strict inequality. The usage of “between” instead of the inequality itself is dictated by the uncertainty of which of those two values is smaller. We cannot just write $f(a)\le k\le f(b)$, since it may happen that $f(b)<f(a)$. To avoid bulky constructions like “if $f(a)\le f(b)$ then $f(a)\le k\le f(b)$, otherwise $f(b)\le k\le f(a)$” the frase “k between $f(a)$ and $f(b)$” is used.

I like your question. However, if you think about it for a moment, you’ll see that neither of your suggested definitions of “between” allow for a terse and correct statement of the Intermediate Value Theorem. That’s because some of the continuous functions $f : [a,b] \rightarrow \mathbb{R}$ are strictly decreasing, hence $f(a)>f(b),$ and hence nothing is “between” $f(a)$ and $f(b)$ according to your suggestions.

To rectify this, I recommend the following:

Definition. Let $V$ denote a real vectorspace. Then given $x \in V$ and $a,b \in V$, we say that $x$ is between $a$ and $b$ iff
either and therefore both of the following hold.

  • $x$ is an element of the convex hull of the subset $\{a,b\}$ of $V$.
  • $x \in a \vee b$, where $a$ and $b$ are viewed as elements of the poset of convex subspaces of $V$ (note that singleton subsets are
    automatically convex.)

We say that $x$ is strictly between $a$ and $b$ iff $x$ is not only
between $a$ and $b$, but also distinct from both $a$ and $b$.

This specializes to $\mathbb{R}$ as follows:

Proposition. Given $x \in \mathbb{R}$ and $a,b \in \mathbb{R}$, we have that $x$ is between $a$ and $b$ iff $x \in [a,b] \cup [b,a]$, and also that $x$ is strictly between $a$ and $b$ iff $x \in (a,b) \cup (b,a)$.

In this language, the intermediate value theorem states:

Intermediate Value Theorem. Given a continuous function $f : [a,b] \rightarrow \mathbb{R}$ and $y$ between $f(a)$ and $f(b)$, we have that $y \in \mathrm{img}(f)$.