Is the space of possible positions of a particle in QM discrete? I recently updated my understanding about quantum mechanics from popular science level to basic undergraduate level.
What surprised me is that for the quantum state of a particle, the wave function for that state using the position of that particle as a basis, is a function defined on the continuum.
I understand now how there can be a discrete space of energy levels of a particle when it is trapped.


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*But when it comes to the position of a particle, does the fact that we represent the state of a particle as a continuous wave function on position space mean that QM states that the position of a particle can potentially be observed anywhere on the continuum?

*similarly, Im not sure how this works for non trapped particles. It seems energy is not quantized there since momentum alao has a wave function defined on the continuum, so does that mean that the space of possible kinetic energy levels for a non trapped particle is not discrete?

*if the answers to these questions are yes, how does this square with my popular science understanding that there is a minimum segment in space, time, and energy levels given by plancks constant?
 A: to answer your questions:
1> Yes, that is the case. 
2> Yes, that is the case also. However, I don't really understand what you mean by "the momentum also has a wavefunction". Do you refer to the Fourier transform of the spatial wavefunction.
3> The segment you refer to is not in constrast to the previous statements. Planck's constant $\hbar$ always comes into play. For example, if you remember the hamiltonian eigenvalues at the harmonic oscillator:
$$E_n=\left(n+\frac{1}{2}\right)\hbar\omega$$
A: In response to your third question, the Plank length might be what you consider the "quantized" space and the Plank time is the "quantized" time. These are fundamental ideas for loop-quantum gravity, but not a result of the Schrodinger equation or quantum mechanics. 
In quantum mechanics, the frequency (and thus energy) of a particle's wavefunction are quantized, but it may be found anywhere in space. The probability of finding it in a region is given by the integral of the amplitude squared of the particle's wavefunction over that region.
