Absolutely Continuous is Continuous

Theorem

Let $I$ be an interval of $\R$.

If a function $f: I \to \R$ is absolutely continuous, then it is continuous on $I$.

Proof

Take any $x \in I$. We will prove that $f$ is continuous at $x$.

In order to do this we will use the $\epsilon$-$\delta$ definition of continuity.

Take any $\epsilon > 0$, and choose the corresponding $\delta > 0$ obtained from the definition of absolute continuity for $f$.

If $y \in I$ is any point such that $\vert y - x \vert < \delta$, then:

$\vert f(y) - f(x) \vert < \epsilon$

as $f$ is absolutely continuous. This is the particular case of just one interval in the definition of absolute continuity.

Hence, $f$ is continuous at $x$.

As $x$ was arbitrary, this proves that $f$ is continuous on all of $I$.

$\blacksquare$