Absolutely Continuous Function is Continuous
Let $I$ be an interval of $\R$.
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 $I$.