# Tangent cone and tangent space

Let $M$ be a set in $\mathbb{R}^n$ such that $M$ is locally a graph of some differentiable function (not necessarily $C^1$ ). Let $p\in \bar M$. We can define a tangent cone $C(M,p)$ as a set of all tangent vectors at point $p$, where tangent vector $v\in\mathbb{R}^n$ is defined by the following property:

(1) $v$ is tangent iff $\exists (p_n)_{n\in\mathbb{N}}\subset M$ : $p_n\rightarrow p, \ \lim_{n \to \infty} \frac{p_n-p}{|p_n-p|}=\frac{v}{|v|}$

How to prove that in this case $C(M,p)$ is a tangent space $T_pM$ in the standard sense (equivalence class of curves) ?

Without some extra conditions on the closure $\overline M$ (stratifications come to mind), there may be trouble. For instance, suppose $M$ is the famous graph of $y=\sin\frac{1}{x}$ for $x>0$, and let $p$ be the origin. As limits $\frac{p_n-p}{|p_n-p|}$ one gets all unit vectors $(a,b)$ with $0<a\le1$. However, there is no curve (even continuous) in $M$ that reaches the origin. If you want that such a curve exists, then you can change the curve for $x\ge1$ into a loop that reaches the origin from the semiplane $x>0$ with some slope you choose, and still the cone has much more vectors.