The Brouwer fixpoint theorem states that any continuous mapping $f$, from a convex, compact set to itself will contain a fixpoint.
The Brouwer algorithm finds these (approximate) fixpoints. But how is a continuous function inputted into this algorithm?
Is it as a black box? If so, would changing this to a discrete set of points change the complexity of the algorithm?
There are several ways to formalize the BROUWER problem. The most intuitive is to have $f$ specified by an arithmetic circuit, as in Etessami and Yannakakis, On the complexity of Nash equilibria and other fixed points. However, this version is not known to lie in PPAD (see this tutorial of Etessami).
To fix this, we can either restrict the circuit to be piecewise linear, as in Etessami and Yannakakis; or ask that the function be Lipschitz with an explicit constant, as in Rubinstein, Settling the complexity of computing approximate two-player Nash equilibria; or use a discretization which corresponds to a (highly restricted) piecewise linear function, as in Daskalakis, Goldberg and Papadimitriou, The complexity of computing a Nash equilibrium.
