We have reversibly switched the resonance of a GaAs-AlAs microcavity in the near-infrared near λ=1300 nm within 300 fs by the electronic Kerr effect. We reveal by pump-probe spectroscopy a remarkable red shift or blue shift of the light confined inside the cavity for small pulse delays, depending on their temporal ordering. The color-converted light is efficiently generated in a broad frequency continuum that differs markedly from the instantaneous cavity resonance in terms of the central frequency and bandwidth. From observations on cavities with different quality factors, we identify the role of the local density of optical states (LDOS) available to the newly generated light frequencies. In particular, we distinguish the effect of the LDOS related to the cavity resonance itself, and the LDOS continuum that leaks in from the vacuum surrounding the cavity. Our new insights provide a unified picture for seemingly disparate results in traditional and nanophotonic nonlinear optics