We conducted the first long-term (60 days), multiwavelength (optical, ultraviolet, and X-ray) simultaneous monitoring of Cen X-4 with daily Swift observations, with the goal of understanding variability in the low mass X-ray binary Cen X-4 during quiescence. We found Cen X-4 to be highly variable in all energy bands on timescales from days to months, with the strongest quiescent variability a factor of 22 drop in the X-ray count rate in only 4 days. The X-ray, UV and optical (V band) emission are correlated on timescales down to less than 110 s. The shape of the correlation is a power law with index γ about 0.2-0.6. The X-ray spectrum is well fitted by a hydrogen NS atmosphere (kT = 59 - 80 eV) and a power law (with spectral index Γ = 1.4 - 2.0), with the spectral shape remaining constant as the flux varies. Both components vary in tandem, with each responsible for about 50% of the total X-ray flux, implying that they are physically linked. We conclude that the X-rays are likely generated by matter accreting down to the NS surface. Moreover, based on the short timescale of the correlation, we also unambiguously demonstrate that the UV emission can not be due to either thermal emission from the stream impact point, or a standard optically thick, geometrically thin disc. The spectral energy distribution shows a small UV emitting region, too hot to arise from the accretion disk, that we identified as a hot spot on the companion star. Therefore, the UV emission is most likely produced by reprocessing from the companion star, indeed the vertical size of the disc is small and can only reprocess a marginal fraction of the X-ray emission. We also found the accretion disc in quiescence to likely be UV faint, with a minimal contribution to the whole UV flux.