Borehole resistivity measurements are routinely employed to measure the electrical properties of rocks penetrated by a well and to quantify the hydrocarbon pore volume of a reservoir. Depending on the degree of geometrical complexity, inversion techniques are often used to estimate layer-by-layer electrical properties from measurements. When used for well geosteering purposes, it becomes essential to invert the measurements into layer-by-layer values of electrical resistivity in real time. We explore the possibility of using deep neural networks (DNNs) to perform rapid inversion of borehole resistivity measurements. Accordingly, we construct a DNN that approximates the following inverse problem: given a set of borehole resistivity measurements, the DNN is designed to deliver a physically reliable and data-consistent piecewise one-dimensional layered model of the surrounding subsurface. Once the DNN is constructed, we can invert borehole measurements in real time. We illustrate the performance of the DNN for inverting logging-while-drilling (LWD) measurements acquired in high-angle wells via synthetic examples. Numerical results are promising, although further work is needed to achieve the accuracy and reliability required by petrophysicists and drillers.