A function with k continuous derivatives is called a C^k function. In order to specify a C^k function on a domain X, the notation C^k(X) is used. The most common C^k space is C^0, the space of continuous functions, whereas C^1 is the space of continuously differentiable functions. Cartan writes humorously that "by 'differentiable, ' we mean of class C^k, with k being as large as necessary." Of course, any smooth function is C^k, and when l>k, then any C^l function is C^k. It is natural to think of a C^k function as being a little bit rough, but the graph of a C^3 function "looks" smooth. Examples of C^k functions are ( left bracketing bar x right bracketing bar )^(k + 1) (for k even) and x^(k + 1) sin(1/x), which do not have a (k + 1)st derivative at 0.

Banach space | calculus | C^∞ function | continuous function | continuously differentiable function | differential equation