The proof of the following theorem
Let $M$ be an o-minimal structure, and let $a,b\in M,A\subseteq dom(M)$. If $a\in acl(b,A)$ and $a\notin acl(A)$, then $b\in acl(A)$.
starts like this:
Suppose $b\in acl(A,a)$ and $b\notin acl (A)$. We have to prove that $a\in acl(A,b)$. In a totally ordered structure, $acl$ coincides with $dcl$, so there is a function $f:M\longrightarrow M$ which is definable over $A$ and sends $a$ to $b$. $f$ is possibly partial, but one can extend it constantly without any additional parameters (so only with parameters in $A$).
Now why is the last sentence true? Suppose for example $M=(\mathbb R,<)$, $A=\varnothing$, in order to make the situation simpler.
Thank you in advance.
Your proposed counterexample doesn't make sense - what is the function $f$ that you're seeking to extend? In fact, the only unary definable function in $(\mathbb{R},<)$ is the identity function, which is already total.
In the general case, let's make sure we're clear about what it means to extend a partial definable function to a total definable function. So suppose $\varphi_f(x,y)$ has the property that for every $a$, there is at most one $b = f(a)$ such that $\varphi_f(a,b)$ holds. Then the domain of $f$ is definable by $\exists z\, \varphi_f(x,z)$. If $\psi_g(x,y)$ also defines a partial function $g$ with domain containing the complement of the domain of $f$, then we can extend $f$ by $g$ to a total function with the formula $$\varphi_f(x,y) \lor (\lnot \exists z\, \varphi_f(x,z)\land \psi_g(x,y)).$$
When the authors write "extend it constantly" it does sound like they mean to extend a partial definable function $f$ by a constant function (defined by the formula $y=a$), and thus they're assuming that the set of parameters $A$ is nonempty, or that the theory has at least one definable element.
But there's no need to make this assumption: the formula $x=y$ defines the identity function, so any partial unary definable function (in a single-sorted theory) can always be extended by the identity to a total definable function.