How do these restrictions on the wavefunction logically arise?

My lecturer notes that the following restrictions on the wavefunction (in addition to it being a solution to the Schrodinger equation) are logical:


i) Continuous (mathematical definition of a continuous function)
  

ii) Single-valued (every x input has precisely one y output)
  

iii) Have a continuous derivative i.e. can define the gradient at every point


Why are these restrictions necessary?


