Absolutely Continuous Real Function 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 $$\vert y - x \vert < \epsilon$$, then
 * $$\vert f(y) - f(x) \vert < \delta,$$

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 $$I$$.