After thinking about it some more. This question is really a generalization of the intermediate value theory. The IVT is really a homotopy theory question, where you are detecting pi_0(R-{0}), in this case, we are detecting pi_1(R^2-{0}).

I guess a more elegant way to see it, is to consider again, the map f: I x I -> S^1, when restricted to the boundary, d(IxI) ~= S^1, we find (from the argument above) that f_d(IxI) : S^1 -> S^1 winds once, but f_IxI gives a homotopy from this map to the constant map, hence a contradiction.

Perhaps i should have been more clear. f(IxI) is a homotopy 'of paths that start at the top of the circle and end at the bottom'.

By 'paths', i mean paths starting at the top of the circle, and ending at the bottom.