If I'm mistaken anywhere here, please correct me (and please don't be mean about it).
One describes that “When one transfers from the outgoing inertial frame to the incoming inertial frame there is a jump discontinuity in the age of the Earth-based twin.”
This doesn’t in any way explain the paradox. What is a “jump discontinuity” and why would an acceleration cause such a thing? There is really no explanation here.
A jump discontinuity is a sudden, instantaneous change. For example the function y = |x|/x has a jump discontinuity at x = 0, where it jumps straight from -1 (when x < 0) to +1 (when x > 1).
You need to understand that speed is relative, but acceleration is not. If we're traveling down the highway and I observe you behind me traveling toward me at 2 miles per hour, you can equally argue that I'm traveling backward toward you at 2 miles per hour. And somebody on the ground can equally argue that you're going 62 and I'm going 60.
But suppose that you see a deer and suddenly hit the brakes. You feel the acceleration, and you cannot argue that you are actually not moving and I am speeding up ahead of you. Note that unless you went from 62 to 0 instantly, this is not a jump discontinuity.
Similarly, symmetry is broken in the twin paradox when one instantaneously turns around by the star; in your example Fun cannot argue that he stood still and Lazy suddenly stopped and turned around. And this is a jump discontinuity.
This new frame of reference observes the earth at a later point in time. If the traveling twin were to slow down, stop, and then accelerate back he would observe the earth aging faster than usual during this acceleration, and light would travel faster than c (this is no contradiction, since light must travel at c only from an inertial frame of reference). When you jump instantaneously from one inertial frame to another, the earth jumps instantaneously from a past to future point (from your frame of reference).
Here's a way to think of age jumping (and relativity of simultaneity): suppose ink stamps are printing a 12-inch ruler onto a belt of paper that is moving at a speed close to that of light. The stamps all press onto the paper at once and produce a 12-inch ruler onto a moving belt of paper that is length contracted, and the ruler will grow longer once the paper stops.
But from the point of view of an ant on the piece of paper moving toward the ink stamps, the length of the ink stamps has shrunk to something less than 12 inches, and so when they push down the ruler is less than twelve inches. When the paper stops, nobody can agree on the length of the ruler. Paradox? No.
What happens from the ant's perspective is that although the ink stamps have shrunk, they don't all hit the paper at the same time. The ones further away hit first, and the ones closer hit later, and since the inks are moving toward you this makes the ruler longer.
From a bystander's point of view, the ant sees further into the future the further he looks ahead, as he sees the further ink stamps hit first.
Similarly, when moving toward the earth you observe it at a more distant point into the future than when you were moving away from it. So the jump in age is just part of relativity, like time dilation and length contraction.