Ant colony optimization for continuous functions
No, discretizing solution space is not necessary

I read page 14 of paper you provided and then went googling.

I found this 2014 paper: A unified ant colony optimization algorithm for continuous optimization that mentioned a bit of history of ACO on continuous function.

Tracing from that, I think the best paper to begin is this 2008 paper Ant colony optimization for continuous domains coauthored by the original creator of ACO, Marco Dorigo, himself.

Quote from page 76 this paper:


  The fundamental idea underlying $ACO_R$ is the shift from using a
  discrete probability distribution to using a continuous one...


That is instead of remembering pheromone value in discrete boxes, you remember in a form of probability distribution function (PDF) that is parameterized for updates. The moving of the ants (solution points) is also continuous.
The ants move random directions across the dimensions of solution space (as opposed to moving from box to box as in the first paper you gave.)

I hope I have answered some of your question and provide some references for further reading without digging too deep in myself.