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A function is continuous on an open interval if the interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval ( − ∞ , + ∞ ) {\displaystyle (-\infty ,+\infty )} (the whole real line ) is often called simply a continuous function; one also says that ...
Definition: We say that the function (resp. set-valued function) f is closable in X × Y if there exists a subset D ⊆ X containing S and a function (resp. set-valued function) F : D → Y whose graph is equal to the closure of the set Gr f in X × Y. Such an F is called a closure of f in X × Y, is denoted by f, and necessarily extends f.
So, if the open mapping theorem holds for ; i.e., is an open mapping, then is continuous and then is continuous (as the composition of continuous maps). For example, the above argument applies if f {\displaystyle f} is a linear operator between Banach spaces with closed graph, or if f {\displaystyle f} is a map with closed graph between compact ...
The usual proof of the closed graph theorem employs the open mapping theorem.It simply uses a general recipe of obtaining the closed graph theorem from the open mapping theorem; see closed graph theorem § Relation to the open mapping theorem (this deduction is formal and does not use linearity; the linearity is needed to appeal to the open mapping theorem which relies on the linearity.)
The Müntz–Szász theorem is a basic result of approximation theory, proved by Herman Müntz in 1914 and Otto Szász (1884–1952) in 1916. Roughly speaking, the theorem shows to what extent the Weierstrass theorem on polynomial approximation can have holes dug into it, by restricting certain coefficients in the polynomials to be zero.
The Heine–Cantor theorem asserts that every continuous function on a compact set is uniformly continuous. In particular, if a function is continuous on a closed bounded interval of the real line, it is uniformly continuous on that interval. The Darboux integrability of continuous functions follows almost immediately from this theorem.
The space C [a, b] of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space C (a, b) of continuous functions on (a, b), for it may contain unbounded functions.
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of n-connectedness generalizes the concepts of path-connectedness and simple connectedness.