If a rational function on an irreducible affine algebraic set is not regular at a point

Question

Let $X\subseteq \mathbb{A}^n$ be an irreducible affine algebraic set and $f\in k(X)$ a rational function on $X$. $f$ is said to be regular at a point $x\in X$ if it can be written in the form $f=P/Q$ with $P,Q\in k[X]$ and $Q(x)\neq 0$. When the ground field is $\mathbb{R}$ or $\mathbb{C}$, is it true that "$f$ is not regular at $x$ $\Rightarrow$ $f$ is not bounded in any neighborhood (with respect to the usual topology on $\mathbb{R}^n$ or $\mathbb{C}^n$) of $x$?

Answer 1

No, the implication is not true:
Let $X\subset \mathbb C^2$ be the curve $y^2=x^3$ and consider the rational function $f=\frac yx$.
It is not regular at the origin $O=(0,0)$ but it is bounded on every disc $D$ centered at $O$ because $f^2=x$ is bounded on $D$. (Same holds after replacing $\mathbb C$ by $\mathbb R$)