I'm reading the companion book of analysis of Korner, My question is about a step in the proof of theorem 7.12 (second order taylor series)
Note: $f_{12}$ means $(f_{2})_{1}$
Theorem
Suppose $\delta > 0$, $B(0, \delta) \subset \mathbb{R}^2$ and that $f: B(0, \delta) \rightarrow R$. If $f_1, f_2, f_{11}, f_{12}, f_{22}$ exist in $B(0, \delta)$ and $f_{11}, f_{12}, f_{22}$ are continuous at $0$ and $f_1(0) = f_2(0) = f_{11}(0) = f_{12}(0) = f_{22}(0) = 0$, then
$$ f(h, k) / (h^2 + k^2) \to 0 $$ as $(h^2 + k^2)^{1/2} \to 0 $
Proof:
If $\epsilon > 0$, the continuity of given partial derivatives at $0$ tell us that we can find a $\delta_1(\epsilon)$ such that $\delta > \delta_1(\epsilon)> 0$ and
$\|f_{11}\|, \|f_{12}\|, \|f_{22}\| \le \epsilon $ if $(h, k) \in B(0, \delta_1(\epsilon))$
Using the mean value inequality:
$$|f_1(h,k) - f_1(h, 0)| \le \epsilon|k|$$
(the proof continue...)
This is my question:
In my understanding the proof is using the fact that $|f_{12}| \le \epsilon$, but we need in this step $|f_{21}| \le \epsilon$ and this is not granted. if this is this correct, why?
In fact, using this theorem he show that if $f_{12}$ exist in $B(0, \delta)$ and $f_{12}$ is continuous at $0$ too, then $f_{12}(0) = f_{21}(0)$