The sketch hopes to connect our informal idea of continuity of a function to a more formal definition. For these three cases, determine if the function is or is not continuous by the epsilon delta definition. That is, given a point, pick an epsilon. For that epsilon, could you find a delta so that the red bit of the function is in the blue epsilon range? Now could you do that for any epsilon? If you can do that for any point, then the function is continuous.
Some people think of a continuous function as one you can draw without lifting your pen. How does the epsilon-delta condition prevent having to lift your pen? Extension: a function is uniformly continuous if for any epsilon you choose, you can find a delta that works for every point. Is case 1 uniformly continuous? More GeoGebra sketches are at mathhombre.blogspot.com.