quantum mechanics - What's the point of Pauli's Exclusion Principle if time and space are continuous?

What does the Pauli Exclusion Principle mean if time and space are continuous?

Assuming time and space are continuous, identical quantum states seem impossible even without the principle. I guess saying something like: the closer the states are the less likely they are to exist, would make sense, but the principle is not usually worded that way, it's usually something along the lines of: two identical fermions cannot occupy the same quantum state

Answer

The other answer shows nicely how one may interpret the Pauli exclusion principle for actual wavefunctions. However, I want to address the underlying confusion here, encapsulated in the statement

If time and space are continuous then identical quantum states are impossible to begin with. in the question.

This assertion is just plainly false. A quantum state is not given by a location in time and space. The often used kets $\lvert x\rangle$ that are "position eigenstates" are not actually admissible quantum states since they are not normalized - they do not belong to the Hilbert space of states. Essentially by assumption, the space of states is separable, i.e. spanned by a countably infinite orthonormal basis.

The states the Pauli exclusion principle is usually used for are not position states, but typically bound states like the states in a hydrogen-like atom, which are states $\lvert n,\ell,m_\ell,s\rangle$ labeled by four discrete quantum numbers. The exclusion principle says now that only one fermion may occupy e.g. the state $\lvert 1,0,0,+1/2\rangle$, and only one may occupy $\lvert 1,0,0,-1/2\rangle$. And then all states at $n=1$ are exhausted, and a third fermion must occupy a state of $n > 1$, i.e. it must occupy a state of higher energy. This is the point of Pauli's principle, which has nothing to do with the discreteness or non-discreteness of space. (In fact, since the solution to the Schrödinger equation is derived as the solution to a differential equation in continuous space, we see that non-discrete space does not forbid "discrete" states.)