Often in my studies (economics) the assumption of a "well-behaved" function will be invoked. I don't exactly know what that entails (I think twice continuously differentiability is one of the requirements), nor do I know why this is necessary (though I imagine the why will depend on each case).
Can someone explain it to me, and if there is an explanation of the why as well, I would be grateful. Thanks!
EDIT: To give one example where the term appears, see this Wikipedia entry for utility functions, which says at one point:
In order to simplify calculations,
various assumptions have been made of
utility functions.CES (constant elasticity of substitution, or
isoelastic) utility
Exponential utility
Quasilinear utility
Homothetic preferencesMost utility functions
used in modeling or theory are
well-behaved. They are usually
monotonic, quasi-concave, continuous
and globally non-satiated.
I might be wrong, but I don't think "well-behaved" means monotonic, quasi-concave, continuous and globally non-satiated. What about twice differentiable?
Best Answer
In the sciences (as opposed to in mathematics) people are often a bit vague about exactly what assumptions they are making about how "well-behaved" things are. The reason for this is that ultimately these theories are made to be put to the test, so why bother worrying about exactly which properties you're assuming when what you care about is functions coming up in real life which are probably going to satisfy all of your assumptions.
This is particularly ubiquitous in physics where it is extremely common to make heuristic assumptions about well-behavior.
Even in mathematics we do this sometimes. When people say something is true for n sufficiently large, they often won't bother writing down exactly how large is sufficiently large as long as it's clear from context how to work it out. Similarly, in an economics paper you could read through the argument and figure out exactly what assumptions they need, but it makes it easier to read to just say "well-behaved."