As discussed in the answer to How can one derive Schrödinger equation?, one should be able to "derive" the Schrodinger equation from the path integral formulation of quantum mechanics.
However, as noted in Feynman's paper (and highlighted in the answer to the previous question), that derivation depends on $$ \psi(x, t + \epsilon) = \int \dfrac{\exp{\dfrac{i}{\hbar} \epsilon L(\bar{x}, \bar{\dot{x}})}}{A} \psi(x + \gamma, t) d \gamma$$ being true up to first order in $\epsilon$, even though the paper only states that the equation "is not exact but is only true in the limit $\epsilon \to 0$." As this equation comes from the the definitions of the path integral that requires inserting more and more intermediate positions, I see why this would be justified in the limit as $\epsilon \to 0$, but don't see what type of argument would make it valid to first order in $\epsilon$.
I suppose if we assume the Schroedinger equation, then the above equation must hold to first order in $\epsilon$. But how can we see that this is true without assuming the Schroedinger equation (in order that we may prove it)?
