quantum mechanics - Can a physical wavefunction be non-smooth (its first derivative is discontinuous)?

Here's an argument that might support the statement that such a non-smooth wavefunction is not physical:

You cannot add a finite number of smooth functions to get a non-smooth function. By fourier expansion theorem, any function can be expressed as a sum of plane waves (which are smooth with respect to spatial dimensions). Hence you must need an infinite number of smooth functions to get a non-smooth function. Now here's the problem. The different plane waves are momentum eigen functions and the non-smooth function is a superposition of these momentum eigen functions. If you now try to calculate the expectation value of the momentum, due to the fact that the momentum eigen values associated with the momentum eigen functions are unbounded from above, the expected value of the momentum could blow up (go to infinity). This is specifically what would make the wavefunction non-physical.

$$\psi(x) = \int \widetilde{\psi}(k) e^{ikx} \mathrm{dk}$$

$$E(k) = \int |\widetilde{\psi}(k)|^2 k \ \mathrm{dk}$$

But for a general smooth function, how do I know whether the fourier coefficients associated with larger and larger momentum eigen modes taper off fast enough for the expected value of the momentum to converge?