Continuous at Point iff Left-Continuous and Right-Continuous

Theorem

Let $A \subseteq \R$ be an open set of real numbers.

Let $f : A \to \R$ be a real function.

Let $x_0 \in A$.

Then:

$f$ is continuous at $x_0$

if and only if:

$f$ is both left-continuous and right-continuous at $x_0$

Proof

Necessary Condition

Suppose $f$ is continuous at $x_0$.

Then, by definition:

$\ds \lim_{x \mathop \to x_0} \map f x = \map f {x_0}$

By Limit iff Limits from Left and Right:

$\ds \lim_{x \mathop \to x_0^-} \map f x = \map f {x_0}$

$\ds \lim_{x \mathop \to x_0^+} \map f x = \map f {x_0}$

Therefore, by definition, $f$ is both left-continuous and right-continuous at $x_0$.

$\Box$

Sufficient Condition

Suppose $f$ is both left-continuous and right-continuous at $x_0$.

Then, by the respective definitions:

$\ds \lim_{x \mathop \to x_0^-} \map f x = \map f {x_0}$

$\ds \lim_{x \mathop \to x_0^+} \map f x = \map f {x_0}$

By Limit iff Limits from Left and Right:

$\ds \lim_{x \mathop \to x_0} \map f x = \map f {x_0}$

Therefore, by definition, $f$ is continuous at $x_0$.

$\blacksquare$