In This Paper We Obtain some Subsets of the Set Ir of Real Numbers on Which Fractional Part Function Is Continuous As a Real-Valued Function of a Real Variable. Every Real Number Can Be Written As the Sum of Its Integral Part and Fractional Part. This Gives Rise to Fractional Part Function As a Real-Valued Function of a Real Variable. This Function Is Helpful In the Study of Dynamics on Circle. some Properties of Such Subsets of Ir Are Obtained.