In one of my exercises on Algebraic Geometry, I showed that the curve $X \subset \mathbb{A}^2$ defined by $x^3-y^2$ is irreducible but not smooth. Furthermore, they ask the following question that I don't understand.
Let $P$ be the point $(0,0)$. Show that there is no pair $(U,f)$ with $P \in U \subset X$ open affine, $f \in \mathcal{O}_X(U)$ and $v_P(f)=1$ (Hint: consider $k[x,y]/m^2$ with $m=(x,y)$)
Note: $v_P(f)$ is defined here as:
Let $X$ be an irreducible curve, $P \in X$ and $f \in K(X)^\times$. If there exists an affine open $U \subset X$ with $P \in U$ such that $f_{|U} \in \mathcal{O}_X(U)$ and $f$ has no zeros on $U - {P}$, then we define: \begin{align*} v_P(f) = dim_k \mathcal{O}_X(U) / (f_{|U}) \end{align*}
I don't know how to show that there is not such a pair $(U,f)$ and how to use the hint.