While the mid-infrared spectral region spans more than 3000cm-1, ultrafast mid-IR spectroscopies are normally limited to the spectral bandwidth that can be generated in optical parametric amplifiers-typically a few hundred cm-1. As such, the spectral coverage in conventional two dimensional infrared (2D IR) spectroscopy captures only about 1% of the full potential 2D mid-IR spectrum. Here, we present 2D IR spectra using a continuum source as both the excitation and probe pulses, thus capturing close to the full 2D IR spectrum. While the continuum pulses span the entire mid-IR range, they are currently too weak to efficiently excite molecular vibrational modes but strong enough to induce electronic responses and excite phonons in semiconductors. We demonstrate the full spectrum 2D IR spectroscopy of the mid-IR bandgap semiconductor indium arsenide with a bandgap at 2855cm-1. The measured response extends far below the bandgap and is due to field-induced band-shifting, causing probe absorption below the bandgap. While the band-shifting induces an instantaneous response that exists only during pulse overlap, the 2D IR spectra reveal additional off-diagonal features that decay on longer timescales. These longer-lived off-diagonal features result from coherent phonons excited via a Raman-like process at specific excitation frequencies. This study illustrates that the full spectrum 2D IR spectroscopy of electronic states in the mid-IR is possible with current continuum pulse technology and is effective in characterizing semiconductor properties.