Question

How do you find a periodic function with no period?

Answer 1

Hint: For such given problems, we need to have a crystal clear concept of what a periodic function is and how a periodic function differs from that of a non-periodic function. Theoretically speaking, a periodic function is one in which the graph of the function repeats itself after regular intervals of time. All other functions are basically non-periodic functions. Mathematically speaking, a function \[f\left( x \right)\] is said to be a periodic function if and only if \[f\left( x+T \right)=f\left( x \right)\] , where ‘T’ is the period. Conversely speaking, if the function \[f\left( x \right)\] is such that \[f\left( x+T \right)=f\left( x \right)\] is true, then \[f\left( x \right)\] is a periodic function with period ‘T’.

Complete step by step solution:
Now we start off with our problem by saying that, we need to find a function which is a periodic function but doesn’t have any period. In other words, we can mathematically as well as theoretically show that a function is periodic and has no period. Consider a constant function such that,
\[f\left( x \right)=c\] , where ‘c’ is any constant. Now for any point in the real axis the function is defined to be equal to ‘c’. Thus we can safely write that for any point other than ‘x’ (say ‘x+T’) the functional value is equal to ‘c’. Hence we can write,
\[f\left( x+T \right)=c\]
When we plot the graph of this constant function we can very clearly observe that the constant function has no period. In other words, we cannot specify a particular interval which repeats after a regular interval of time.

Note:
In solving these types of problems we need to have a clear cut idea of what a periodic and a non-periodic function is. We also need to understand how a periodic function gets repeated after regular intervals of time and how it is mathematically represented. Any constant function defined is a periodic function with no period, because we cannot find a point or an interval in which it keeps on repeating.