The man's escape strategy described in the post was discovered in 1952 by Besicovitch. It has a kind of discontinuity because, quoting the post, "There are 2 perpendicular paths. Choose the one closest to the center, or if the two paths are equally close, then either one is fine." In simple terms, the man will eventually make a wrong turn (with probability 1) unless he can instantaneously measure distances with perfect accuracy.

Surprisingly, for every continuous man strategy, there is a continuous lion strategy that can catch the man by time T where T is the disc radius divided by the lion's speed.

Restricting to continuous strategies in some other lion-man games actually leads to other paradoxes such as both lion and man having a "winning" strategy. The shallow resolution of the paradox is that two such winning strategy cannot actually be played against each other.

http://arxiv.org/abs/0909.2524

The deeper resolution is that even continuous strategies can be unphysical if they allow for information to travel at infinite speed (e.g., if the man is modelled as knowing the lion's current speed and velocity, special relativity notwithstanding). I'm not aware a proof in the literature, but presumably continuity of strategies plus an information speed limit will avoid the above paradoxes. (Continuous-time game theory is still very immature compared to discrete-time game theory.)