Wavefunction and Nodes

[Kyle Sheu 1C]

Hi folks! Can anyone explain why the wavefunction must pass through zero and not simply approach zero in order for there to be a node?

With the Born interpretation of a wavefunction, my understanding is that there is a zero probability density whenever the wavefunction is at zero, regardless of whether the function touches zero or passes through zero, so I'm curious to know why the function must go through zero. Would it perhaps be that a sign change (positive to negative or vice versa) in the wavefunction model is a necessary characteristic of nodes?

Thank you!

[Chem_Mod]

Yes