Let $x=(x_i)_{1\leq i\leq n}$ be the "actions" of $n$ players, where $x_i\in[0,1]$ is determined by player $i$ seeking to maximize its objective function $\pi_i(x)=\pi_i(x_i,x_{-i})\geq 0$.
Assume that \begin{align} \text{(i)}\quad &\pi_i(x)=0 \;\text{if}\; x=0. \\ \text{(ii)}\quad &\pi_i(x)\to\infty \;\text{if}\; x_{-i}=0 \;\text{and}\; x_i\to 0. \end{align} In this case, what kind of a point is $x=0$? It seems like it cannot be an equilibrium, because player $i$ has an incentive to chose $x_i>0$, but at the same time it has an incentive to choose $x_i$ as small as possible. Is $x=0$ some kind of a "unstable" or "singular" equilibrium?
It is not an equilibrium because the best response to $x_{-i}=0$ is $x_i =\varepsilon$ for a small value of $\varepsilon$. Recall that a Nash equilibrium is a fixed point where the action of any player $i$ is the best response to the actions of the rest of the players. In addition, you cannot talk about stability because this is not an equilibrium