Uniformly Continuous Function is Continuous

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

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

Proof
Take any $$x \in I$$, and let us prove that $$f$$ is continuous at $$x$$ by using the $$\epsilon$$-$$\delta$$ definition of continuity.

For any $$\epsilon > 0$$, take the $$\delta > 0$$ given by the definition of uniform continuity for $$f$$.

Then, for any $$y \in I$$ such that $$\vert x - y \vert < \epsilon$$, as $$f$$ is uniformly continuous we have that
 * $$\vert f(x) - f(y) \vert < \delta$$.

This proves that $$f$$ is continuous at $$x$$.

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

Notes on the proof
The above proof just states that the condition of being continuous on $$I$$ is already explicitly included in the condition of being uniformly continuous on $$I$$.