the action is not a well defined function

The action of moving from q_a to q_b in time t is not a well defined function

for example, consider the harmonic oscillator

if q_a= q_b, then only for t being the multiple of the harmonic oscillator period will the particle return to its initial position

Therefore, for such kind of q_a and q_b, the action is ill-defined for an arbitrary t

but people are always taking the derivatives of the action to the time!!!!!!