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:

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