Concept of periodic function

Periodic function $f$

A number $T>0$ is called a period of $f$ if for every $x$ in the domain, whenever $x$ lies in $D(f)$ so does $x+T$ , and $f(x+T)=f(x).$ The smallest positive $T$ among all such numbers (if it exists) is called the least positive period $T_{\min}$ — in graph problems people often say simply “the period”. A constant $f(x)\equiv C$ satisfies the condition for any $T>0$ — it is usually excluded from intuitive “wave” examples; for non-trivial sine and tangent, periods are the key to reading the graph along the whole axis.

$T$ is invariant under shift along the argument axis: the graph coincides with itself when translated by $T$

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