I was looking through my lecture notes, and I noticed the definition of a function:
One of the three components of a function, $f :X \to Y$ is as follows: '$f$ is a $\mathbf{well-defined}$ rule that assigns a unique elements $f(x) \in Y$ to each $x \in X.$'
It then goes on to say: '$f$ is a $\mathbf{well-defined}$ or $\mathbf{unambiguous}$ if and only if, $f(x)=f(x')$ whenever $x=x'.$'
Surely this is incorrect, as this means that the definition does not include the constant function?
As if $f : X \to Y$ and $f(x)=a$, then for $x, x' \in X $ and $x \neq x'$, $f(x) =f(x')$?
Actually a constant function is not a counterexample to the definition. If $P$ implies $Q$, then the negation of $P$ does not necessarily imply the negation of $Q$. It is that the negation of $Q$ implies that of $P$. Think of this example: Suppose we agree that if $x$ is a person then $x$ will die. From this it does not follow that if $x$ is not a person then $x$ will not die. But it follows from the implication that if $x$ will not die then $x$ is not a person.
According to the definition that confused you, it is trivial for a constant function to satisfy the condition required; the functional value of a constant function at every point of its domain is the same single point of its codomain.