Intro to Periodic Data

Say we have data in the form $(x, t)$, where $x$ is some measured parameter and $t$ is the time of measurement. Additionally, assume the data is periodic with fundamental period $\tau$.

Now assume there exists $T > \tau$ and a data point $(x^*, t^*)$ such that $(x^*, t^* + T)$ is also a data point. Must $(x^*, t^* + 2T)$ also be a data point?