Neighbor says: >I'd like to have an explanation of why we cannot, and then after hearing what's the reason, we will see how to proceed.
Well, mathematicians don't always have "explanations", so they offer up logic instead. Logic can go where explnations fear to tread...
Anyway, in this case, (assuming we want those two "drops of glue" to be *distinct* points both before and after the lines are joined,) then we can bring them close together but either:
1. There is always some small distance left between them. or 2. There is no distance between them! But then they lose their distinctness -- the two points become identified as one single point (which is OK, I guess, but I'm playing along with *your* requirement that the two line segments are *disjoint* - I presume you want them disjoint before and after separation and joining, no?)
So we have case 1 then. But if there is any distance between them, no matter how small, then we haven't really joined them into a single continuous unbroken line.