8. This problem is about the definition of periodic function. Weassume you already know intuitively what periodic means, and now wewant a formal definition. For simplicity, we will restrictourselves to functions with domain R. A naive (but incorrect)definition of periodic function with period T is

   f (x + T ) = f (x)

   Without accompanying words, this is not a good definitionbecause it does not introduce the variables x and T and it does notexplain their role. For which values of x and T does the abovedefinition have to be valid?

   Here is an attempt at a definition, with various ways tocomplete it:Definition. Let f be a function with domain (??, ?). Wesay that

   f is periodic when...
   (a) Foreveryx?(??,?)andforeveryT >0,f(x+T)=f(x).
   (b) For every x ? (??,?) there exists T > 0 such that f(x+T) =f(x). (c) There exists T > 0 such that x ? (??, ?) =? f (x + T )= f (x). (d) There exists T > 0 such that for every x ? (??,?),f(x+T) = f(x). (e) For every T > 0 there exists x ? (??,?) suchthat f(x+T) = f(x).

9. One or more of the above are valid ways to complete thedefinition of periodic function. Identify which ones are correctand which ones are wrong. For any property which is wrong, show itby giving an example of a function which satisfies the property butis not periodic, or an example of a function which is periodic butdoes not satisfy the property. It is okay to give your examples asequations or graphs.