Which of the following describes a periodic function?

A periodic function is defined as one that repeats its values at regular intervals over its domain. This means that for some constant ( T ), if ( f(x) ) is a periodic function, then ( f(x + T) = f(x) ) for all values of ( x ). Common examples of periodic functions include sine and cosine functions, which repeat their values every ( 2\pi ) radians.

Understanding periodic functions is crucial in various applications across mathematics, physics, and engineering, as they model repetitive phenomena such as sound waves, seasons, and other cyclical patterns. The requirement for regular intervals is what distinguishes these functions from others that may behave differently over their ranges.

While other options may describe characteristics of functions, they do not encompass the concept of periodicity. A function that has only one value is constant, a function that does not repeat varies continuously or has distinct values without periodicity, and functions with singularities refer to points at which they may not be well-defined, like poles or discontinuities. These different characteristics are important in function theory but do not relate directly to the periodic nature of functions.