We use LCOGT observations (MJD 59434-59600) with a total exposure time of ~50hr and a median cadence of 0.5d to measure the interband time delays (with respect to u) in the g, r, and i continua of a highly variable active galactic nucleus (AGN), 6dFGS gJ022550.0-060145. We also calculate the expected time delays of the X-ray reprocessing of a static Shakura-Sunyaev disk according to the source's luminosity and virial black hole mass; the two parameters are measured from the optical spectrum of our spectroscopic observation via the Lijiang 2.4m telescope. It is found that the ratio of the measured time delays to the predicted ones is 2.6_-1.3_^+1.3^. With optical light curves (MJD53650-59880) from our new LCOGT and archival Zwicky Transient Facility, Pan-STARRS, Catalina Sky Survey, and ATLAS observations, and Wide-field Infrared Survey Explorer (WISE) data (MJD 55214-59055), we also measured time delays between WISE W1/W2 and the optical emission. W1 and W2 have time delays (with respect to V), 9.6_-1.6_^+2.9^x10^2^d and 1.18_-0.10_^+0.13^x10^3^d in the rest frame, respectively; hence, the dusty torus of 6dFGS gJ022550.0-060145 should be compact. The time delays of the W1 and W2 bands are higher than the dusty torus size-luminosity relationship of Lyu+ (2019ApJ...886...33L). By comparing the infrared and optical variability amplitude, we find that the dust-covering factors of the W1 and W2 emission regions are 0.7 and 0.6, respectively. Future broad emission-line reverberation mapping of this target and the results of this work enable us to determine the sizes of the AGN main components simultaneously.