# Equivalence of Definitions of Continuous Real Function at Point

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## Theorem

The following definitions of the concept of Continuous Real Function at Point are equivalent:

### Definition 1

$f$ is continuous at $x$ if and only if the limit $\ds \lim_{y \mathop \to x} \map f y$ exists and:

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

### Definition 2

$f$ is continuous at $x$ if and only if the limit $\ds \lim_{y \mathop \to x} \map f y$ exists and:

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

## Proof

### $(1)$ 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 $2$.

$\Box$

### $(2)$ 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 $1$.

$\blacksquare$