Lead climbing without belayers (SPOILERS)

Hello,

Just watched this film and found it to be quite good. But a technical aspect of the climbing was nagging me. When Martin and Roger are solo climbing the mountain at the end they have no partners for the belaying yet they have each have a rope and are lead climbing. So my question is, is it possible to solo climb with a rope for the entire height of a mountain? Can you solo multi-pitch climb with a rope? I was under the impression that a partner is necessary.

I know that most climbing films tend to disregard the reality of climbing for the drama, but this film seemed as though it was aiming for realism.