Been enjoying Physics Forums and this is my first question so before I ask I should I applaud everyone for the helpful friendly atmosphere. I'm trying to understand the concepts of continuity and connectedness. I have no set theory background nor physics backgrounds so I just read online for guidance so please keep answers to a relatively conceptual level. I understand continuity as it applies to mathematics in regard to epsilon-delta. Also, as I understand it, continuity when it applies to topologies seems to be a basic tenet. Space-time seems to be described as a four-dimensional, smooth, connected Lorentzian manifold. On this type of a manifold it seems to be that connectedness and path-connectedness are the same. I understand that this allows for comparing frames of reference and movement of energy and particles. It seems to me in my very uneducated guess that this allows for differentiability (which at least in mathematics means you can figure out instantaneous rates of change). My question is, in discrete space-time models such as LQG, they say that it also talks of a differentiable manifold. On the other hand they say that according to LQG space is not continuous. My understanding is that if something is differentiable it is necessarily continuous. Certainly my lay understanding is leading my a-stray.. can anyone help me fill in the gaps to my thinking? MUCH APPRECIATION.