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In mathematics, the term undefined refers to a value, function, or other expression that cannot be assigned a meaning within a specific formal system. [1]Attempting to assign or use an undefined value within a particular formal system, may produce contradictory or meaningless results within that system.
Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.
Removable singularity – Undefined point on a holomorphic function which can be made regular; Mathematical singularity – Point where a function, a curve or another mathematical object does not behave regularly; Extension by continuity – Property of topological space
A function which never returns has an undefined value because the value can never be observed. Such functions are formally assigned the bottom type, which has no values. Examples fall into two categories: Functions which loop forever. This may arise deliberately, or as a result of a search for something which will never be found.
[3] [4] Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient norm is equal to zero (or undefined). [ 5 ] This sort of definition extends to differentiable maps between R m {\displaystyle \mathbb {R} ^{m}} and R n , {\displaystyle \mathbb {R} ^{n},} a critical point ...
A function that is not well defined is not the same as a function that is undefined. For example, if f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} , then even though f ( 0 ) {\displaystyle f(0)} is undefined, this does not mean that the function is not well defined; rather, 0 is not in the domain of f {\displaystyle f} .
Undefined behavior, computer code whose behavior is not specified under certain conditions; Undefined value, a condition where an expression does not have a correct value; Undefined variable; Undefined, a function or variable lacking a declaration; Undefined, a variable lacking initialization
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.