For instance, an integrable function may be less smooth (in some appropriate sense) than a continuous function, which may be less smooth than a differentiable function, which may be less smooth than a twice differentiable function, and so on. Some insight will reveal that this process of grouping functions into classes is an attempt to categorize functions with respect to how "smooth" or "well-behaved" they are.
Examples of such classes are the continuous functions, the differentiable functions, the integrable functions, etc. One divides these functions into different classes depending on their properties. Elementary calculus may be described as a study of real-valued functions on the real line.
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