General fractional integral operators do not preserve periodicity

If a periodic real function is differentiable, its derivative is also periodic with the same period. The primitive of a periodic function is periodic if the mean value of the function, over a period, is zero. It is well known that the fractional integral of a nontrivial periodic function cannot be periodic, that is, a nonconstant periodic function cannot have periodic fractional integral, unless it is zero, of any period. Recently R. Garrappa et al. proved that, for a fractional integral with a Sonine kernel, the action on periodic functions does not preserve the periodicity of any period. In this note we show the same nonperiodic nature for a general fractional integral operator. • General fractional integral operators do not preserve the periodicity of nonconstant periodic functions. • Extends the nonperiodicity result known for Sonine-kernel fractional integrals (Garrappa et al.) to a general setting. • Clarifies implications for periodic solutions in fractional models.