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Mathematicians (and those in related sciences) very frequently speak of whether a
mathematical object -- a
number, a
function, a
set, a space of one sort or another -- is '''"well-behaved"''' or not. While the term has no fixed formal definition, it can have fairly precise meaning within a given context.
In pure mathematics, "well-behaved" objects are those that can be proved or analyzed by elegant means to have elegant properties.
In both pure and applied mathematics, (
optimization,
numerical integration, or
mathematical physics, for example,) ''well-behaved'' also means not violating any of the assumptions needed for the successful application of whatever analysis is being discussed.
The opposite case is usually labelled
pathological.
Generally,
*
Lebesgue-integrable functions are better-behaved than general functions in
calculus.
*
Riemann-integrable functions are better-behaved than Lebesgue-Integrable functions in
calculus.
*
Continuous functions are better-behaved than
Riemann-integrable functions on compact sets in
calculus.
*
Continuous functions are better-behaved than discontinuous ones in
topology.
*
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.
*
Smooth functions are better-behaved than general differentiable functions.
*
Analytic functions are better-behaved than general smooth functions
*
Euclidean space is better-behaved than
non-Euclidean geometry.
*Attractive
fixed points are better-behaved than repulsive fixed points.
*
Fields are better-behaved than
skew fields.
*
Hausdorff topologies are better-behaved than those in arbitrary
general topology.
*
Separable field extensions are better-behaved than
non-separable ones.
*
Borel sets are better-behaved than arbitrary
sets of
real numbers.
*Spaces with
integer dimension are better-behaved than spaces with
fractal dimension.
*Spaces with
finite dimension are better-behaved than spaces with
infinite dimension in
linear algebra.
