Infinite compositions of analytic functions

Analytic functions are functions that have derivatives at every point in their domain. That means that they don't break into pieces but instead have a smooth curve or line. You can create an infinite number of analytic functions when you combine two or more of these functions together. For example, you can add two smooth functions together to create a new, more complex smooth function that is made of pieces of the two old functions. It's like making a new toy by putting pieces of two other toys together. Every time you combine two more functions the new toy will be different.