Finding period of any type of function

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That explains it all. Thank you everybody.
 
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period of tan pi x = 1 as the graph for x = -1/2 to 1/2 repeats.
period of x - [ x ] = 1 as the graph repeats for [n , n-1).
Hence, the period of the given function is 1.

AGain for Sin (2 pi x ) + x - [x] also the period is 1, for a similar reason.