What is a wave function and what are the requirements for it to be well-behaved, i.e. for it to properly represent physical reality?

Hint: In quantum physics, the wave function can be used to depict a particle's wave characteristics. As a result, the quantum state of a particle can be characterised by its wave function.
The probability of an element's quantum state as a function of position, momentum, time, and spin is defined using this interpretation of the wave function. The Greek letter Psi $\left( \psi \right)$ is used to signify it.

Complete step by step solution:
In quantum physics, a wave function is a mathematical representation of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude from which probabilities for the potential outcomes of system observations can be calculated.
The wave function that is single valued, continuous, and finite is known as a well behaved wave function. There are various characteristics that a wave function must meet in order to be considered a well behaved (or full) wave function:-
1. Because there can only be one probability value at any given position, the wave function must be single valued in each given coordinate \[\left( {x,{\text{ }}y,{\text{ }}z} \right)\] .
2. In order for the second derivative $\left( {\dfrac{{{\delta ^2}y}}{{\delta {x^2}}}} \right)$ to exist and perform properly, the wave function must be continuous.
3. Finally, the wave function must be finite in order to obtain a normalised wave function, and $\left( \psi \right)$ must also be integrable.

Note:
It is important to emphasize, however, that the wave function itself has no physical importance. However, its proportionate value of \[{\psi ^2}\] at a given moment and location has physical significance.