A function with continuous derivatives is called a function. In order to specify a function on a domain , the notation is used. The most common space is , the space of continuous functions, whereas is the space of continuously differentiable functions. Cartan (1977, p. 327) writes humorously that "by 'differentiable,' we mean of class , with being as large as necessary."
Of course, any smooth function is , and when , then any function is . It is natural to think of a function as being a little bit rough, but the graph of a function "looks" smooth.
Examples of functions are (for even) and , which do not have a st derivative at 0.
The notion of function may be restricted to those whose first derivatives are bounded functions. The reason for this restriction is that the set of functions has a norm which makes it a Banach space,