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A function f from X to Y. The set of points in the red oval X is the domain of f. Graph of the real-valued square root function, f(x) = √ x, whose domain consists of all nonnegative real numbers. In mathematics, the domain of a function is the set of inputs accepted by the function.
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 ...
In complex analysis, a complex domain (or simply domain) is any connected open subset of the complex plane C. For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function.
The space is said to be path-connected (or pathwise connected or -connected) if there is exactly one path-component. For non-empty spaces, this is equivalent to the statement that there is a path joining any two points in X {\displaystyle X} .
The Cauchy's integral theorem states that if is a simply connected open subset of the complex plane, and : is a holomorphic function, then has an antiderivative on , and the value of every line integral in with integrand depends only on the end points and of the path, and can be computed as () ().
In convex analysis, a branch of mathematics, the effective domain extends of the domain of a function defined for functions that take values in the extended real number line [,] = {}. In convex analysis and variational analysis , a point at which some given extended real -valued function is minimized is typically sought, where such a point is ...
This concept is often contrasted with uniform convergence.To say that = means that {| () |:} =, where is the common domain of and , and stands for the supremum.That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent.
It is important to specify the domain and codomain of each function, since by changing these, functions which appear to be the same may have different properties. Injective and surjective (bijective) The identity function id X for every non-empty set X , and thus specifically R → R : x ↦ x . {\displaystyle \mathbf {R} \to \mathbf {R} :x ...