Mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

s (and those in related sciences) very frequently speak of whether a mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 object — a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

, a set, a space of one sort or another — is "well-behaved" or not. The term has no fixed formal definition, and is dependent on mathematical interests, fashion, and taste. To ensure that an object is "well-behaved" mathematicians introduce further axioms to narrow down the domain of study. This has the benefit of making analysis easier, but cuts down on the generality of any conclusions reached. Concepts like non-Euclidean geometry were once considered ill-behaved, but are now common objects of study.

In both pure and applied mathematics (optimization

Optimization (mathematics)

In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....

, numerical integration

Numerical integration

In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...

, or mathematical physics

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

, for example), well-behaved also means not violating any assumptions needed to successfully apply whatever analysis is being discussed.

The opposite case is usually labeled pathological

Pathological (mathematics)

In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved....

. It is not unusual to have situations in which most cases (in terms of cardinality) are pathological, but the pathological cases will not arise in practice unless constructed deliberately.

Despite the list below, in practice "well-behaved" is almost always applied in an absolute sense.

• In calculus
  Calculus
  Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
  
  :
  • Analytic function
    Analytic function
    In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
    
    s are better-behaved than general smooth function
    Smooth function
    In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
    
    s.
  • Smooth functions are better-behaved than general differentiable functions.
  • Continuous differentiable functions are better-behaved than general continuous functions. The larger the number of times the function can be differentiated, the more well-behaved it is.
  • Continuous function
    Continuous function
    In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
    
    s are better-behaved than Riemann-integrable functions on compact sets.
  • Riemann-integrable functions are better-behaved than Lebesgue-integrable
    Lebesgue integration
    In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
    
     functions.
  • Lebesgue-integrable functions are better-behaved than general functions.
• In topology
  Topology
  Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
  
  , continuous functions are better-behaved than discontinuous ones.
  • Euclidean space
    Euclidean space
    In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
    
     is better-behaved than non-Euclidean geometry
    Non-Euclidean geometry
    Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
    
    .
  • Attractive fixed points
    Fixed point (mathematics)
    In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
    
     are better-behaved than repulsive fixed points.
  • Hausdorff topologies are better-behaved than those in arbitrary general topology
    General topology
    In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
    
    .
  • Borel set
    Borel set
    In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
    
    s are better-behaved than arbitrary sets of real number
    Real number
    In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
    
    s.
  • Spaces with integer
    Integer
    The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
    
     dimension are better-behaved than spaces with fractal dimension
    Fractal dimension
    In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
    
    .
  • Finite-dimensional
    Dimension (vector space)
    In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...
    
     vector spaces are better-behaved than infinite-dimensional ones.
• In abstract algebra
  Abstract algebra
  Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
  
  :
  • Fields
    Field (mathematics)
    In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
    
     are better-behaved than skew field or general rings
    Ring (mathematics)
    In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
    
    .
  • Separable field extension
    Field extension
    In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
    
    s are better-behaved than non-separable ones.
  • Normed division algebra
    Normed division algebra
    In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || · || satisfying the following property:\|xy\| = \|x\| \|y\| for all x and y in A....
    
    s are better-behaved than general composition algebras.