<p>Is it possible to connect two pieces of 1.75 mm filament end to end, with no change in width? I am asking the question because I am interested in creating a multi-filament feeder to a single extruder, and I am curious about the process of changing filament while the 3-d printer extruder continues uninterrupted. My current best-guess at the optimal solution is to someone 'cut' one end of the filament and 'melt' it to the end of another filament.</p>

You'd have to ensure that the joining portion of the two filaments do not "bloom" or increase in diameter, which would happen if unconstrained at the melting and joining time. Alignment is also critical, otherwise you have a varying diameter from one color to the next at the point of join.
There's an item on ebay which is precision drilled and has precision machined mating surfaces to ensure alignment of the filament. The filament is extended through the two components, heated (in the case of the video, with a match) and the two metal parts are pushed together. The joining of the metal parts also cuts away the excess bulge of the melted filament, ensuring correct diameter.
In the case of the above item, the joining devices have to be threaded with the filament prior to the joining process and then have to be threaded off the filament by pushing them the entire length of the filament.
This would be okay if one were joining short lengths only. Consider how much fun it isn't to have to slide these two pieces over an entire full spool of filament.
A split device which would enable one to un-latch or otherwise open a clamshell to release the filament would require much more challenging machining to achieve the necessary precision, which is probably why we don't see such a product.