Validity of the derivation of time-energy uncertainty principle? Background
The top voted answer to this question seems to make some assumptions I'm uncertain about (to say the least): What is $\Delta t$ in the time-energy uncertainty principle?
Assumptions


*

*The uncertainty principle is usually a statement about the measurement (while it is true they are applicable during the unitary process as well. They usually are not used in that context).

*Since they usually the measurement is considered a discontinuous process is he considering an interpretation where there is no measurement such as many worlds?


Question
Is there a version of this derivation (compatible with the discontinuous measurement interpretations of quantum mechanics) where one is allowed to use the calculus in the way he does (both integration and differentiation)?
P.S: I have already commented on this.
 A: The WP summary of the Mandeshtam-Tamm relation  is, for an observable $\hat B$,
$$
\sigma_E ~~~\frac{\sigma_B}{\left| \frac{\mathrm{d}\langle \hat B \rangle}{\mathrm{d}t}\right |} \ge \frac{\hbar}{2} ~~,
$$
where the second factor on the l.h.s., with dimensions of time, is a lifetime of the state ψ with respect to the hermitean observable $\hat B$. Roughly, the time interval (Δt) after which the expectation value ⟨$\hat B$⟩  changes appreciably. 
For a stationary state, the drift rate of ⟨$\hat B$⟩ goes to zero, and the variance of energy goes to 0 as well, as it should.
This is all in standard QM, unitarily evolving, with or without measurements. You may do any and all measurements discontinuous, delirious, expialidocious, whatever, and plot your results, but you must be talking about the same state ψ all the time. The distribution in B will have a variance, which is what is under discussion. 
(Heuristically,  a state  ψ that only exists for a short time cannot have a definite energy. To have a definite energy, the frequency of the state must be defined accurately, and this requires the state to  persist for many cycles, the reciprocal of the required accuracy. In spectroscopy, excited states have a finite lifetime. By  above, they do not have a definite energy, and, each time they decay, the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the natural linewidth. Fast-decaying states have a broad linewidth, while slow-decaying states have a narrow linewidth.)
I have not fully appreciated your misgivings, but they seem to me to also apply to the standard Δx Δp uncertainty principle: A pure state will have corresponding distributions for x and p with nontrivial variances, computable through standard continuous QM, which your measurements will probe. 
