I'm reading a proof of Hadamard-Perron theorem from Katok's Introduction to the Modern Theory of Dynamical Systems. I'm having problems with the following part.
Let $\varphi:\mathbb{R}^k\to\mathbb{R}^{n-k}$ be a Lipschitz function with Lipschitz constant = $\gamma$. For $x\in\mathbb{R}^k$ define $$ \Delta_y\varphi := \frac{(y,\varphi(y))-(x,\varphi(x))}{||(y,\varphi(y))-(x,\varphi(x))||} \text{ for } x \neq y, $$ $$ t_x\varphi := \{v \in T_x\mathbb{R}^n : \exists \{x_n\}_{n \in \mathbb{N}} \text{ such that } \lim_{n\to\infty}x_n=x \land \lim_{n\to \infty}\Delta_{x_n}\varphi=v \}. $$ Then by a tangent set to $\varphi$ at $x$ we mean the set $$\tau_x\varphi := \bigcup_{v\in t_x\varphi}\mathbb{R} v$$
Now we have:
Lemma. $\varphi$ is differentiable at $x$ $\iff$ $\tau_x\varphi$ is a $k$ - dimensional plane,
where by a $k$ - dimensional plane we mean a $k$ - dimensional subspace of $\mathbb{R}^n$.
There's no proof of this in the book. I know how to prove it in one dimensional case, where $\varphi:\mathbb{R}\to\mathbb{R}$, because it's all about the limit-definition of a derivative. I tried to adopt one dimensional case to multidimensional by proving that if the tangent set was "something more" than a $k$ - dimensional plane, then one of the partial derivatives wouldn't exist but with no success.
Also, I couldn't find anything helpful here. If I missed something, please redirect me.
P.S. I'm not looking for a solution, hints would work better for me :)
Many thanks for any help!