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In Real Analysis, while we are constructing the Real Numbers Axiomatically, we (in some books) define one important Axiom, Axiom of Continuity, which goes like this :

"If $A, B\subseteq\mathbb{R}$ and the set $A$ lies left from the set $B$, that is : $\forall a \in A, \forall b \in B \implies a \leq b$, then $\exists c \in\mathbb{R} :\forall a \in A, \forall b \in B \implies a \leq c \leq b$.

My question is : How pictorially (geometrically intuitively) can we draw this Axiom in a straight line and understand the concept of this Axiom?

Edit : The position of $c$ ?

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    $\begingroup$ For a geometric interpretation, if every element of $a$ is bounded above by any element of $b$, then there exists a separator between the sets. (In fact, any element in $[\sup A,\inf B]$ would work). $\endgroup$
    – Rushabh Mehta
    Commented May 9, 2021 at 19:01

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enter image description here

In the above diagram, the red point and line segment is the set $A$.

The green point and green line segment is the set $B$.

Every element of $A$ (red) on the left of the elements of $B$ (green).

The axiom claims that we can find a purple dot, that is on the right of the set $A$ and on the left of set $B$.

The choice of $c$ need not be unique, for example, I could have let $c$ to be any value in $[2, 2.5]=[\sup A, \inf B]$. (Of course, the notion of supremum and infimum might not have been developed)

Consider another pair of $A$ and $B$

$A= \{x: x^2 < 2, x>0, x\in \mathbb{Q} \}$ and $B= \{x: x^2 > 2, x>0, x \in \mathbb{Q}\}$, the axiom promises that we can find a real element between $A$ and $B$.

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