In physics and engineering we see periodic phenomena such as vibrations, the motion of the tide, planetary, and alternating current (AC).
- The condition “$x+T$ lies in the domain of $f$ whenever $x$ lies in the domain of $f$” does not say that $f$ needs to be defined for all $x$. For example,
- The period of a periodic function is not unqiue. In fact, if $f$ is periodic with $T$, it is also periodic with periods $2T$, $3T,\dots$ , or $-T$ and $-2T,\dots$ because
\] and so on. In general:
- A constant function $f(x)\equiv3$ is periodic, and every number $T$ is its period
\] Therefore, there is no smallest period, and hence $f$ does not have a fundamental period.