The perimeter (which is something very close to the derivative) is not continuous. If you think about it a bit, you can easily change a shape in a way that gre

The perimeter (which is something very close to the derivative) is not continuous. If you think about it a bit, you can easily change a shape in a way that greatly increases its perimeter but leaves it's area (almost) unaffected. Which means that just because two shapes are extremely close in shape, they can be very different in perimeter. What is interesting is that it is semi continuous. The limit (this isn't quite correct) of the series is larger than the point. As the limit of the series is 2, but the actual distance is sqrt(2), which is smaller. So if you calculate the limit of the staircase perimeter you only know that it is larger or equal to the perimeter of the limit. >I suppose the answer is that it is not in fact ever a triangle — just a thing with a fat edge. It absolutely is a triangle. Because it converges in shape.