Many natural functions are continuous (or almost continuous except at a few points), so it pays to study continuous functions in general (rather than repeating the same analysis on each continuous function one by one). Theoretically, continuous functions have nice properties, such as the substitution principle (useful when finding limits) and intermediate value theorem. They also provide the simplest example of functions that can be integrated. They are relevant not just to calculus, but other subjects as well such as topology and economics.
