The question may be too vague, but ultimately in search of various (counter)examples or theorems to exhibit the following:


Do continuous families $t\mapsto G\_t$ of "games" (say each $G\_t$ is a stable finite game with unique Nash equilibrium) have that the Nash equilibrium varies continuously?; or a fixed point?; or discontinuously? Here the family may be a 1-parameter $t\in[0,1]$ or simply an infinitesimal deformation $t\in(-\varepsilon,\varepsilon)^n$ of a given game $G\_0(x\_1,\ldots,x\_n)$ depending on $n$ hyperparameters.



Continuity of Nash equilibrium for a family of games
Usually, one looks at the correspondence (set-valued map) that associates with each game its set of Nash equilibria. Under many conditions, this correspondence is [upper hemicontinuous](https://en.wikipedia.org/wiki/Hemicontinuity). For example, if one fixes finite action spaces for a fixed finite set of players, then the correspondence from payoff functions, represented as points in a Euclidean space whose dimension is the number of pure strategy profiles times the number of players, is upper hemicontinuous.


A function can be identified with a single-valued correspondence, and the function is continuous at a point if and only if the corresponding correspondence is upper hemicontinuous. So if you restrict yourself to parameters for which there is a unique Nash equilibrium, it will be a continuous function of these parameters.


