Differentiability Classes
A function ƒ is said to be continuously differentiable if the derivative ƒ′(x) exists, and is itself a continuous function. Though 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 x≠0,
which has no limit as x → 0. Nevertheless, Darboux's theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem.
Sometimes continuously differentiable functions are said to be of class C1. A function is of class C2 if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of class Ck if the first k derivatives ƒ′(x), ƒ″(x), ..., ƒ(k)(x) all exist and are continuous. If derivatives f(n) exist for all positive integers n, the function is smooth or, equivalently, of class C∞.
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