Can I create a continuous curve using numerical discrete data? In this case, I'm looking at the area of Sierpinski triangles which decrease at a rate of $\frac{3}{4}$ per reiteration of the fractal. Since the area is represented in a geometric progression wouldn't an exponential curve fit the infinite data set as the number of re-iterations goes to infinity?
However, I've heard people say that since the number of re-iterations is an integer value, then the curve cannot be continuous.
How can I understand where a continuous curve applies to integer values in a geometric progression?
It's true that the data is discrete, rather than continuous. It can be modeled with a continuous function, however. In particular, the function
$$A(x) = \frac{\sqrt{3}}{4} \left(\frac{3}{4}\right)^x$$
fits the data quite well at the integers. Here's an image (generated on Observable) illustrating the fit: