The spectral range of femtosecond time-resolved infrared spectroscopy is limited by the bandwidth of mid-IR pulses (100~400 cm-1) generated from the combination of Ti:Sapphire amplifier, Optical Parametric Amplifier (OPA), and Difference Frequency Generation (DFG). To overcome this limitation, we implement a compact continuum mid-IR source producing ultrafast pulses that span the frequency range from 1000 to 4200 cm-1 (from 10 to 2.4 μm), which utilize the mixing of fundamental, second-harmonic, and third-harmonic of 800 nm pulse in the air. After building an IR spectrometer with continuum IR and a monochromator, we found that the distortion of the measured IR spectrum originated from the contamination of higher-order diffraction. We used bandpass filters to eliminate the higher-order contributions and correct the measured IR spectrum. We further characterized the spectral properties of fundamental, second-harmonic, and third-harmonic fields after the plasmonic filamentation process, which helps to improve the efficiency of the continuum IR generation. Using the generated continuum IR pulses, we measured the IR absorption spectrum of a water-benzonitrile mixture, which was found to be consistent with the spectrum obtained with a commercial FT-IR spectrometer. The present work will be useful for the efficient generation of continuum IR pulses for IR pump-probe and two-dimensional IR spectroscopy experiments in the future.