How curves are represented in a computer
A curve can be quantized (also known as rasterized) to a certain precision after which it is discrete, and the issue is gone. If you see a picture that contains a curve on a modern computer, there's a good chance it already has been rasterized (e.g. PNG or jpeg).

However, it is possible to represent curves (and other primitives) directly. Then you're looking at vector graphics. In most implementations of vector graphics only a couple specific subsets of curves are supported, most often circles, ellipses and Bézier curves. Especially the latter is the most common powerful generic way of storing curves.

It is possible to describe an arbitrary curve through some mathematical framework and storing the mathematical description of the curve. But such a generalized approach is rarely done for computer graphics.