Left-Hand and Right-Hand Differentiable Function is Continuous

Theorem
Let $f$ be a real function defined on an interval $I$.

Let $a$ be a point in $I$ where $f$ is left- and right-hand differentiable.

Then $f$ is continuous at $a$.

Proof
By Left-Hand Differentiable Function is Left-Continuous, $f$ is left-continuous at $a$.

By Right-Hand Differentiable Function is Right-Continuous, $f$ is right-continuous at $a$.

By Limit iff Limits from Left and Right, $f$ is continuous at $a$ since $f$ is left- and right-continuous at $a$.