Right-Hand Differentiable Function is Right-Continuous

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

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

Then $f$ is right-continuous at $a$.

Proof
By hypothesis, $f'_+ \left({a}\right)$ exists.

First we note that $a$ cannot be the right hand end point of $I$ because values in $I$ greater than $a$ need to exist for $f'_+ \left({a}\right)$ to exist.

We form the following expression:


 * $\displaystyle \lim_{x \mathop \to a^+} \left({f \left({x}\right) − f \left({a}\right)}\right)$

We need to show that it is defined and to find its value.

We find:

Note that this proves that $\displaystyle \lim_{x \mathop \to a^+} \left({f \left({x}\right) − f \left({a}\right)}\right)$ exists.

We continue by manipulating the result above:

which means that $f$ is right-continuous at $a$.