is discontinuous at the origin since it fluctuates infinitely many times between 1 and -1.b. If for f from [-1,1] to the reals, sin(f(x)) is continuous (over the reals) then f must be continuous.

I think f(x)=1/x (restricted to [-1,1]) may be a counterexample to this but one problem is it is not defined at 0 which is in [-1,1], is it ok to use it anyway.

I think the statement is true because of algebra of continuity.