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I was recently reading Griffith Introduction to Quantum Mechanics, and I stuck upon a following sentence:

"...but $\Psi$ must go to zero as $x$ does to $\pm\infty$ - otherwise the wave function wound not be normalizable."

The author also added a footer: "A good mathematician can supply you with pathological counterexamples, but they do not arise in physics (...)".

Can anybody give such a counterexample?

Best, Nick

I was recently reading Griffiths' Introduction to Quantum Mechanics, and I stuck upon a following sentence:

but $\Psi$ must go to zero as $x$ goes to $\pm\infty$ - otherwise the wave function would not be normalizable.

The author also added a footer: "A good mathematician can supply you with pathological counterexamples, but they do not arise in physics (...)".

Can anybody give such a counterexample?