Since high-pass filters correspond to differentiators, and low-pass filters correspond to integrators, does this imply that a band-pass has no associated function?

No. Remember that the integration/differentiation functions are frequency-dependent. A low-pass filter integrates at high $\omega$ (where it also attenuates) only; a high-pass filter differentiates at low $\omega$ (where it also attenuates). A band-pass filter will act like an integrator at high frequency and differentiator at low frequency.