Existence and Equality of the Reversed Mixed Second Partial Derivative

Let $n$ be a natural number, let $\mathbb{R}$ be the real numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^{n}$, let $f:U\to\mathbb{R}$, let $a\in U$, and let $i$ and $p$ satisfy $1\le i\le n$ and $1\le p\le n$, index ranges using the order on the natural numbers. Assume the following.

1. The partial derivative of $f$ with respect to the $i$th variable exists at every point of $U$; let $\partial_i f:U\to\mathbb{R}$ be the function whose value at $y\in U$ is that partial derivative at $y$.

2. The partial derivative of $f$ with respect to the $p$th variable exists at every point of $U$; let $\partial_p f:U\to\mathbb{R}$ be the function whose value at $y\in U$ is that partial derivative at $y$.

3. The partial derivative of $\partial_p f$ with respect to the $i$th variable exists at every point of $U$; let $\partial_i\partial_p f:U\to\mathbb{R}$ be the function whose value at $y\in U$ is that partial derivative at $y$, this being the iterated partial derivative notation of clause 4 of C^k Maps on a Euclidean Open Set. Assume moreover that $\partial_i\partial_p f$ is continuous at $a$.

Then the partial derivative of $\partial_i f$ with respect to the $p$th variable exists at $a$, and its value $\partial_p\partial_i f(a)$ satisfies

$$\partial_p\partial_i f(a)=\partial_i\partial_p f(a).$$