Fractional part function and properties

Solve the following equations (where [*] denotes greatest integer function and {*} represent fractional part function and sgn represents signum function)

The number of solution of the equation sgn({x})=|1-x| is/are (where {*} represent fractional part function and sgn represent signum function)

Property-3, Modulus based || Greatest integer Function based || Fractional part function based question , Discussion Exercise

{ . } denotes the fractional part function and [.] denotes the greatest integer function. Now, match the following lists:

If F(x)=(sinpi[x])/({x}) then F(x) is (where {.} denotes fractional part function and [.] denotes greatest integer function and sgn(x) is a signum function)

The function f(x)=[x]+sqrt({x}), where [.] denotes the greatest Integer function and {.} denotes the fractional part function respectively,is discontinuous at

The number of integers in the range of the function f(x)=(x-[x])/(1+{x}) x(where [.] and {.} denote the integral part and fractional part functions respectively is

Solve the equation [x]+{-x}=2x, (where [.l and {} represents greatest integer function fractional part function respectively)

Write the function f(x) ={sinx} where {.} denotes the fractional part function) in piecewise definition.