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, $\map {f'_+} a$ 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 $\map {f'_+} a$ to exist.

We form the following expression:


 * $\ds \lim_{x \mathop \to a^+} \paren {\map f x - \map f a}$

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

We find:

Note that this proves that $\ds \lim_{x \mathop \to a^+} \paren {\map f x - \map f a}$ exists.

We continue by manipulating the result above:

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