Equivalence of Definitions of Continuous Real Function at Point

$(1)$ implies $(2)$
Let $f$ be a Continuous Real Function at Point by definition $1$.

Then by definition:
 * $\ds \lim_{y \mathop \to x} \map f y = \map f x$

We have that:
 * $\ds x = \lim_{y \mathop \to x} y$

and so it follows that:
 * $\ds \map f x = \map f {\lim_{y \mathop \to x} y}$

That is:
 * $\ds \lim_{y \mathop \to x} \map f y = \map f {\lim_{y \mathop \to x} y}$

Thus $f$ is a Continuous Real Function at Point by definition $2$.

$(2)$ implies $(1)$
Let $f$ be a Continuous Real Function at Point by definition $2$.

Then by definition:
 * $\ds \lim_{y \mathop \to x} \map f y = \map f {\lim_{y \mathop \to x} y}$

We have that:
 * $\ds \lim_{y \mathop \to x} y = x$.

Thus:
 * $\ds \map f {\lim_{y \mathop \to x} y} = \map f x$.

That is:
 * $\ds \lim_{y \mathop \to x} \map f y = \map f x$

Thus $f$ is a Continuous Real Function at Point by definition $1$.