where the proper time is invariant why change (differential) in proper time $d\tau$ is not zero? $\Delta \tau=\tau_f-\tau_i$ as i know. $d(invariant)=0$ note to comment: action $S=-m_oc^2\int_C d\tau$
Invariance always applies between some set of conditions. Things can be invariant in time, invariant over changes in position or over changes of coordinate systems and so on. In relativity things described as "invariant" without a descriptive clause are general things that all observers can agree upon. The claim that "proper time is invariant", means that the proper time between two events (four points in space time) is something that all observers can agree upon, not that the measured proper time is zero.
something that all observers can agree upon like speed of light but this does not means that the measured in speed of light is not zero $dc\not= 0$,
ok guys i got the point: invariance is not a synonym for being a constant. thanks