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In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.
For example, [8] the function (,) = | |, regarded as a complex function with imaginary part identically zero, has both partial derivatives at (,) = (,), and it moreover satisfies the Cauchy–Riemann equations at that point, but it is not differentiable in the sense of real functions (of several variables), and so the first condition, that of ...
For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous. [1] In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem.
This is because that function, although continuous, is not differentiable at x = 0. The derivative of f changes its sign at x = 0, but without attaining the value 0. The theorem cannot be applied to this function because it does not satisfy the condition that the function must be differentiable for every x in the open interval.
Some authors, such as [13] refer to functions satisfying this inequality as elliptic functions. An equivalent condition is the following: [14] () + () + ‖ ‖ It is not necessary for a function to be differentiable in order to be strongly convex.
However, this function is not continuously differentiable. A smooth function that is not analytic. The function = {, < is continuous, but not differentiable at x = 0, so it is of class C 0, but not of class C 1.
Most functions that occur in practice have derivatives at all points or almost every point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points. [14] Under mild conditions (for example, if the function is a monotone or a Lipschitz function), this is true. However, in 1872 ...
The converse is not true for real functions; in fact, in a certain sense, the real analytic functions are sparse compared to all real infinitely differentiable functions. For the complex numbers, the converse does hold, and in fact any function differentiable once on an open set is analytic on that set (see "analyticity and differentiability ...