Monotonicity of wavefunction

[manish1012]

Lecture 10 mentions this.
1. All eigenfunctions cannot be accepted as wave functions. Any
wave function to be acceptable
should follow the following
properties.
(a) Continuous
(b) Differentiable
(c) Monotonic
(d) Square integrable

I think a monotonous function cannot be normalised/square-integrated hence cannot be accepted as a wavefunction. Also wavefunction for particle in a box is non monotonous.But 1. mentions the contrary.

[schrodinger]

I think they meant one value cannot be mapped to more than one value. basically, the function definition. the term monotonic was wrongly used