What makes a function well-behaved in physics?

[sachi]

We are asked to explain what is meant by a "well behaved function" which represents a physical phenomenon. I know that it has to be continuous, and differentiable at all points (therefore we need the gradient to be continuous), but I'm not sure if derivatives of all orders need to be continuous, although this seems intuitively correct.

 

[Meir Achuz]

"WBF" depends on the problem.
In QM, it generally means that u" is finite (for a smooth potential), but u" and higher derivatives can be discontinuous, and usually are.
If the pot is a delta function, then a "well behaved wave function" can have a discontinous first derivative.