Continuously Differentiable Real Function at Removable Discontinuity/Corollary

Theorem
Let $f : \R \to \R$ be a real function.

Let $a \in \R$ be real number.

Let $f$ be $n - 1$ times continuously differentiable in $\R$ and $n$ times continuously differentiable in $\R \setminus \set a$.

Suppose that $a$ is a removable discontinuity of $f^{\paren n}$.

That is, suppose the limit $\ds \lim_{x \mathop \to a} \map {f^{\paren n}} x$ exists.

Then $f$ is $n$ times continuously differentiable at $a$.

Proof
Let $y = f^{\paren {n - 1}}$.

Then $y' = f^{\paren n}$.

The result follows from Continuously Differentiable Real Function at Removable Singularity.