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The definition of a wave function demands continuity and differentiability so that it can satisfy the Schrödinger Equation. My question is whether this assumption is necessary for reality. Does probability density being positive and normalizable demands the wave function to be differentiable or continuous?

We just want $$\int |Ψ(x)|^2\;dx$$

Does all $\Psi(x)$ that satisfy this criterion fall in the differetiable category? A governing differential equation like the Schrödinger Equation lets us predict/understand the full behavior of the system with our knowledge of boundary conditions. Can a system exist without a governing equation like we know it as all differential equations stand with the continuity and differentiablity assumption, otherwise it is useless. Is it possible for a physical system to be governed by something other than a differential equation? Some new mathematical construct that doesn't need this assumptions?

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What we want us not just $$ \int|\Psi (x)|^2\;dx $$

For example $$ p=\int Ψ^*(x)(- i \hbar)\; \text{grad} Ψ(x)\;dx $$

If the wave function is not differeciable $\text{grad}$ will come into a terrible result

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