Left-Hand Differentiable Function is Left-Continuous

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

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

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

Proof
By hypothesis, $\map {f'_-} a$ exists.

First we note that $a$ cannot be the left-hand end point of $I$ because values in $I$ less than $a$ need to exist for $\map {f'_-} a$ to exist.

We form the following expression:


 * $\displaystyle \lim_{x \mathop \to a^-} \paren {\map f x − \map f a}$

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

We find:

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

We continue by manipulating the result above:

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