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Closed graph theorems are of particular interest in functional analysis where there are many theorems giving conditions under which a linear map with a closed graph is necessarily continuous. If f : X → Y is a function between topological spaces whose graph is closed in X × Y and if Y is a compact space then f : X → Y is continuous. [4]
Closed graph theorem [5] — If : is a map from a topological space into a Hausdorff space, then the graph of is closed if : is continuous. The converse is true when Y {\displaystyle Y} is compact .
Precisely, the theorem states that a linear operator between two Banach spaces is continuous if and only if the graph of the operator is closed (such an operator is called a closed linear operator; see also closed graph property). An important question in functional analysis is whether a given linear operator is continuous (or bounded).
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 ...
This is illustrated in the diagram, where the two black oriented circles are labelled with the corresponding value of the argument of the logarithm used in z 3 ⁄ 4 and (3 − z) 1/4. We will use the contour shown in green in the diagram. To do this we must compute the value of f(z) along the line segments just above and just below the cut.
For uniformly continuous functions, for each positive real number > there is a positive real number > such that when we draw a rectangle around each point of the graph with a width slightly less than and a height slightly less than , the graph lies completely inside the height of the rectangle.
Pavel Urysohn. In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem or Urysohn-Brouwer lemma [1]) states that any real-valued, continuous function on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.
The function in example 1, a removable discontinuity. Consider the piecewise function = {< = >. The point = is a removable discontinuity.For this kind of discontinuity: The one-sided limit from the negative direction: = and the one-sided limit from the positive direction: + = + at both exist, are finite, and are equal to = = +.