Search results
Results From The WOW.Com Content Network
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 ...
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 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 ...
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.
If a function is differentiable on an open interval and its derivative is bounded, then the function is uniformly continuous on that interval. Every Lipschitz continuous map between two metric spaces is uniformly continuous. More generally, every Hölder continuous function is uniformly continuous.