Search results
Results from the WOW.Com Content Network
In summary, a set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half-open interval. [4] [5] A degenerate interval is any set consisting of a single real number (i.e., an interval of the form [a, a]). [6] Some authors include the empty set in this definition.
The definition of uniform continuity appears earlier in the work of Bolzano where he also proved that continuous functions on an open interval do not need to be uniformly continuous. In addition he also states that a continuous function on a closed interval is uniformly continuous, but he does not give a complete proof. [1]
In calculus, a function defined on a subset of the real numbers with real values is called monotonic if it is either entirely non-decreasing, or entirely non-increasing. [2] That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.
By definition U c (f) is an open subset of (a, b), so can be written as a disjoint union of at most countably many open intervals I k = (a k, b k). Let J k be an interval with closure in I k and ℓ(J k) = ℓ(I k)/2. By compactness, there are finitely many open intervals of the form (s, t) covering the closure of J k. On the other hand, it is ...
A function: between two topological spaces and is continuous if the preimage of every open set in is open in . [8] The function : is called open if the image of every open set in is open in . An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals.
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 ...
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
A convex function of one real variable defined on some open interval is continuous on . admits left and right derivatives, and these are monotonically non-decreasing. In addition, the left derivative is left-continuous and the right-derivative is right-continuous.