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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.
The Weierstrass function has historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. [1]
A Lipschitz function g : R → R is absolutely continuous and therefore is differentiable almost everywhere, that is, differentiable at every point outside a set of Lebesgue measure zero. Its derivative is essentially bounded in magnitude by the Lipschitz constant, and for a < b , the difference g ( b ) − g ( a ) is equal to the integral of ...
Product rule: For two differentiable functions f and g, () = +. An operation d with these two properties is known in abstract algebra as a derivation . They imply the power rule d ( f n ) = n f n − 1 d f {\displaystyle d(f^{n})=nf^{n-1}df} In addition, various forms of the chain rule hold, in increasing level of generality: [ 12 ]
A function of a real variable is differentiable at a point of its domain, if its domain contains an open interval containing , and the limit = (+) exists. [2] This means that, for every positive real number , there exists a positive real number such that, for every such that | | < and then (+) is defined, and | (+) | <, where the vertical bars denote the absolute value.
If a function is semi-differentiable at a point a, it implies that it is continuous at a. The indicator function 1 [0,∞) is right differentiable at every real a, but discontinuous at zero (note that this indicator function is not left differentiable at zero).
Alberto Calderón proved the more general fact that if Ω is an open bounded set in R n then every function in the Sobolev space W 1,p (Ω) is differentiable almost everywhere, provided that p > n. [9] Calderón's theorem is a relatively direct corollary of the Lebesgue differentiation theorem and Sobolev embedding theorem.
Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if for every x 0 {\displaystyle x_{0}} in its domain , its Taylor series about x 0 {\displaystyle x_{0}} converges to the function in some neighborhood ...