= Solution
The exponent is $iS_N/\hbar$, where
$$
S_N=\sum_{j=0}^{N-1}\epsilon\,\frac m2
\left(\frac{x_{j+1}-x_j}{\epsilon}\right)^2
$$
is the classical <action> of the piecewise-linear interpolating path. For smooth paths its limit is $S=\int_0^T m\dot x^2\,dt/2$. The discretization defines the quantum <integral>; it does not assert that typical quantum paths are differentiable. Weighting histories by their <action> phase is the central formulation of the original path-integral paper.<Space-Time Approach to Non-Relativistic Quantum Mechanics by Richard Feynman>[1948, abstract: amplitudes from histories weighted by the exponential of classical action in units of reduced Planck's constant]{ref}
When <action> differences on the experimental scales are large compared with $\hbar$, nearby nonstationary phases cancel. <stationary phase> selects $\delta S=0$ at fixed endpoints, hence $m\ddot x=0$. The surviving free classical trajectory is $x_{\rm cl}(t)=x_0+(x_T-x_0)t/T$, with <action> $m(x_T-x_0)^2/(2T)$. This is the semiclassical mechanism; it does not mean each individual path becomes a classical path.
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