of class C1. Example: Finitely-times Differentiable (C) For each even integer , the function is continuous and times differentiable at all . At , however, is not times differentiable, so is of class C, but not of class C where . Example: Differentiable But Not Continuously Differentiable (not C1) The function is differentiable, with derivative Because oscillates as → 0, is not continuous at zero. Therefore, is differentiable but not of class C1. Example: Differentiable But Not Lipschitz Continuous The function is differentiable but its derivative is unbounded on a compact set. Therefore, is an example of a function that is differentiable but not locally Lipschitz continuous. Example: Analytic (C) The exponential function is analytic, and hence falls into the class Cω. The trigonometric functions are also analytic wherever they are defined as they are linear combinations of complex exponential functions and . Example: Smooth (C) but not Analytic (C) The bump function is smooth, so of class C∞, but it is not analytic at , and hence is not of class Cω. The function is an example of a smooth function with compact support. Multivariate differentiability classes A function defined on an open set of is said to be of class on , for a positive integer , if all partial derivatives exist and are continuous, for every non-negative integers, such that , and every . Equivalently, is of class on if the -th order Fréchet derivative of exists and is continuous at every point of . The function is said to be of class or if it is continuous on . Functions of class are also said to be continuously differentiable. A function , defined on an open set of , is said to be of class on , for a positive integer , if all of its components are of class , where are the natural projections defined by . It is said to be of class or if it is continuous, or equivalently, if all components are continuous, on . The space of Ck functions Let be an open subset of the real line. The set of all real-valued functions defined on is a Fréchet vector space, with the countable family of seminorms where varies over an increasing sequence of compact sets whose union is , and . The set of functions over also forms a Fréchet space. One uses the same seminorms as above, except