Really nice explanation! I'd also add that it's not enough that p is continuous (only looks at a finite number of bits for each input) -- it must also be uniformly continuous (the number of bits it looks at can't become arbitrarily high for arbitrary inputs). For example, if you exclude the sequence of all ones 111111... from the Cantor set, you can write this function p which always returns false:

    function p(bits) {
      n = 0
      while bits(n) == 1 {
        n++
      }
      return false
    }

This function would loop forever on the all-ones sequence. But since that sequence was excluded, p is a total function which looks at more and more bits depending on how many leading ones there are. While each particular input only demands a finite number of bits, there's no finite number of bits which lets you search the entire space. The compactness of the Cantor set ensures that all continuous functions are also uniformly continuous, but if you remove points from the set, it can stop being compact, and the magic spell can be broken!