We describe a phase-coherent multifrequency lock-in measurement technique that uses the inverse Fourier transform to reconstruct the nonlinear current-voltage characteristic of a nanoscale junction. The method provides separation of the galvanic and displacement currents in the junction and easy cancellation of the parasitic displacement current from the measurement leads. These two features allow us to overcome traditional limitations imposed by the low conductance of the junction and the high capacitance of the leads, thus providing an increase in measurement speed of several orders of magnitude. We demonstrate the method in the context of conductive atomic force microscopy, acquiring current-voltage characteristics at every pixel while scanning at standard imaging speed.