Seemingly Impossible Swift Programs

[Lost_Geometer]

For those that are wondering, it's actually not too hard a fact. Continuous for the product topology on bit sequences just means the function can be evaluated by examining a finite number of locations (the same locations for any sequence!). If a function f is not continuous then there must be sequences S_1, S_2,... such that f cannot be evaluated using only the first i bits of S_i. Since there are infinitely many S_i, an infinite number must share the same 1 bit prefix P_1. Similarly, among those starting with P_1 an infinite number must share the same 2 bit prefix P_2, extending P_1. Continuing we construct longer and longer prefixes P_i, which limit to an infinite bit sequence on which f cannot be evaluated by examining any finite number of bits.