continuous function; there exist functions that are differentiable but not continuously differentiable (an example is given in the section Differentiability classes)...

differentiable functions. If a complex function is differentiable just once on an open set, it is both infinitely differentiable and analytic on that set [citation...

that the value of the right sub-function is used in this position. For a piecewise-defined function to be differentiable on a given interval in its domain...

mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily...

versions of the inverse function theorem for holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces...

Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere...

and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally...

mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each...

that interval. If a function is differentiable and convex then it is also continuously differentiable. A differentiable function of one variable is convex...

zero (note that this indicator function is not left differentiable at zero). If a real-valued, differentiable function f, defined on an interval I of...