Note the Physics
This Illumination deals with position vs. time graphs that are made up of straight-line segments, with sudden changes in slope at the times where the segments are joined. The slope, and thus the velocity, is constant during each segment, but undefined at the junction of two segments. In a graph of real motion the slope of the segments would not change suddenly because an object cannot instantaneously change its velocity -- that would require infinitely large acceleration and force. Consider these graphs as an approximation of a real motion in which the object changes its velocity in a very short, but non-zero amount of time.