Take a Cantor Set, which is composed of an infinite number of points. Consider turning those points into dots and letting a Pacman eat them:
As our Pacman eats the dots, he gets heavier. Imagine that his weight after eats all the dots is 1. Letís graph his weight with time. Obviously, it will not increase gradually, but will instead go up like this:
This is called the Devilís Staircase. In this picture, we colored the area below it to make it more visual.
ITíS NOT A FRACTAL!
Although the Devilís Staircase is formed using a fractal, it is itself not a fractal. Consider calculating the length of this figure. Since there are no diagonal lines in the staircase, in order to get from the lower-left corner to the upper-right corner you have to go up 1 and to the right 1 using vertical and horizontal lines. The total length will thus be 2 at all levels of magnification. The length of fractals, however, must increase with magnification. If you know the geometric method, this will tell you that the dimension of the figure is 1, which is not higher than its topological dimension. According to the formal definition, the Devilís Staircase is not fractal! However, it has self-similarity and many other properties that true fractals have. Because of this, we should really try not sticking to any single definition of a fractal.