point is a meagre set in the space of all continuous functions. Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function. Differentiability classes A function is said to be if the derivative exists and is itself a continuous function. Although the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity. For example, the function is differentiable at 0, since exists. However, for differentiation rules imply which has no limit as Thus, this example shows the existence of a function that is differentiable but not continuously differentiable (i.e., the derivative is not a continuous function). Nevertheless, Darboux's theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem. Similarly to how continuous functions are said to be of continuously differentiable functions are sometimes said to be of A function is of if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of if the first derivatives all exist and are continuous. If derivatives exist for all positive integers the function is smooth or equivalently, of Differentiability in higher dimensions A function of several real variables is said to be differentiable at a point if there exists a linear map such that If a function is differentiable at , then all of the partial derivatives exist at , and the linear map is given by the Jacobian matrix, an n × m matrix in this case. A similar formulation of the higher-dimensional derivative is provided by the fundamental increment lemma found in single-variable calculus. If all the partial derivatives of a function exist in a neighborhood of a point and are continuous at the point , then the function is differentiable at that point . However, the existence of the partial derivatives (or even of all the directional derivatives) does not guarantee that a function is differentiable at a point. For example, the function defined by is not differentiable at , but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function is not differentiable at , but again all of the partial derivatives and directional derivatives exist. Differentiability in complex analysis In complex