I'm reading a theorem about the symmetry of second partial derivatives in Principles of Mathematical Analysis by Prof. Rudin:
Theorem 9.41: Suppose $f$ is defined in an open set $E \subseteq \mathbb R^{2},$ suppose that $D_{1} f$, $D_{21} f$, and $D_{2} f$ exist at every point of $E$, and $D_{21} f$ is continuous at some point $(a, b) \in E$. Then $D_{12} f$ exists at $(a, b)$ and $$ \left(D_{12} f\right)(a, b)=\left(D_{21} f\right)(a, b) $$
Before this theorem, the author defines second partial derivatives as follows:
From my understanding, to define $D_{21} f (a)$, we must have $D_1 f$ exists in some neighborhood of $a$. As such, $D_{21} f$ exists at every point of $E$ implies that $D_{1} f$ exists at every point of $E$.
My question: Is the hypothesis that $D_{1} f$ exists at every point of $E$ is redundant in the theorem 9.41?
Thank you for your clarification?