Equivalence of Definitions of Continuous Real Function at Point

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 $1$.

$\blacksquare$

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.1$ Limits and Continuity