Local uniform continuity characterization for C_f on bv_p(ℝ) when p>1

Ascertain necessary and sufficient conditions on f: ℝ→ℝ for the composition operator C_f: bv_p(ℝ)→bv_p(ℝ) to be locally uniformly continuous when p>1, and determine in particular whether this property is equivalent to f being continuously differentiable.

Background

The paper proves that for bv_1(ℝ), C_f is locally uniformly continuous if and only if f is C^1. In the functional BV_p case, an analogous characterization holds for p>1. It is unknown whether a similar equivalence extends to the sequence space bv_p(ℝ) for p>1, where unbounded sequences complicate the analysis.