Continuity of all composition operators on BV_p[a,b]

Establish whether every composition operator C_f: BV_p[a,b]→BV_p[a,b] is continuous for a fixed p∈[1,+∞).

Background

In BV_p[a,b], acting conditions (e.g., the Josephy theorem) are well-understood, and for p=1 there are classical results ensuring continuity of composition operators. However, the general continuity question for arbitrary p remains unresolved in the functional BV_p setting, unlike the sequence space bv_p setting addressed earlier in the paper.