Cutting sets of continuous functions on the unit interval

For a function f:[0,1]→R, we consider the set E(f) of points at which f cuts real axis. Given f:[0,1]→R and Cantor D⊂[0,1] with {0,1}⊂D, obtain conditions equivalent to conjunction f∈C[0,1] (or f∈C∞[0,1]) D⊂E(f). This generalizes some ideas Zabeti. We observe that, if is continuous, then closed nowhere dense subset f−1[{0}]. Additionally, Intf−1[{0}]=0̸, each x∈{0,1}∩E(f) an accumulation point E(f). Our main result states for F⊂[0,1] x∈{0,1}∩F being F, there exists f∈C∞[0,1] such that F=E(f)=f−1[{0}].