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The Weierstrass function has been 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]
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.
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 ...
If an absolutely continuous function is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely continuous. [5] Every absolutely continuous function (over a compact interval) is uniformly continuous and, therefore, continuous. Every (globally) Lipschitz-continuous function is absolutely continuous. [6] If f ...
A function is continuous on a semi-open or a closed interval; if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the ...
Let C 0 (X, R) be the space of real-valued continuous functions on X that vanish at infinity; that is, a continuous function f is in C 0 (X, R) if, for every ε > 0, there exists a compact set K ⊂ X such that | f | < ε on X \ K. Again, C 0 (X, R) is a Banach algebra with the supremum norm.
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.