Limit of Composite Function/Counterexample

Counterexample to Limit of Composite Function is Image of Limit

Let $f$ and $g$ be the real functions defined as:

$\ds \map g x$

$=$

$\ds \eta$

$\ds \map f y$

$=$

$\ds \begin {cases} y_1 & : y = \eta \\ y_2 & : y \ne \eta \end{cases}$

for $\eta, y_1, y_2 \in \R$.

Then:

$\ds \lim_{y \mathop \to \eta} \map f y = y_2$

and:

$\ds \lim_{x \mathop \to \xi} \map g x = \eta$

But we have:

$\forall x \in \R: \map f {\map g x} = y_1$

and so:

$\ds \lim_{x \mathop \to \xi} \map f {\map g x} = y_1$

Hence it is not true that:

$\ds \lim_{x \mathop \to \xi} \map f {\map g x} = \lim_{y \mathop \to \eta} \map f y$