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The concept of the virtual function solves the following problem: In object-oriented programming, when a derived class inherits from a base class, an object of the derived class may be referred to via a pointer or reference of the base class type instead of the derived class type.
Diagram of diamond inheritance, a problem that virtual inheritance is trying to solve.. Virtual inheritance is a C++ technique that ensures only one copy of a base class ' s member variables are inherited by grandchild derived classes.
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.
A virtual call requires at least an extra indexed dereference and sometimes a "fixup" addition, compared to a non-virtual call, which is simply a jump to a compiled-in pointer. Therefore, calling virtual functions is inherently slower than calling non-virtual functions.
A class containing a pure virtual function is called an abstract class. Objects cannot be created from an abstract class; they can only be derived from. Any derived class inherits the virtual function as pure and must provide a non-pure definition of it (and all other pure virtual functions) before objects of the derived class can be created.
In various contexts, things are often described as "virtual" when they share important functional aspects with other things (real or imagined) that are or would be described as "more real". These include the following:
Fundamental sequences arise in some settings of definitions of large countable ordinals, definitions of hierarchies of fast-growing functions, and proof theory.Bachmann defined a hierarchy of functions in 1950, providing a system of names for ordinals up to what is now known as the Bachmann–Howard ordinal, by defining fundamental sequences for namable ordinals below . [9]
In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup H ≤ G {\displaystyle H\leq G} such that H has property P.