Specifying the domain of a function is an essential part of its definition. There may be cases when the domain is taken for granted but, luckily, in this case it is quite clearly stated: f(x)*f(f(x))=1 for all real x.

Our only conclusion is that f cannot be continuous.

We can reduce its domain and use it as part of a new composite function, but making a continuous link between (999,1/999) and (1000,999), will mean that not all the real values of x in this interval will satisfy the equation f(x)*f(f(x))=1.