A quantum point contact (QPC), a narrow region separating two wider electron reservoirs, is the standard building block of sub-micron devices, such as quantum dots - small boxes of electrons, and qubits - the proposed basic elements of quantum computers. As a function of its width, the conductance through a QPC changes in integer steps of G0 = 2e2/h, signalling the quantization of its transverse modes. Such measurements also reveal an additional shoulder at a value around 0.7 × G0 which has become known as the 0.7 anomaly. Recently it has been suggested that this phenomenon can be explained if one invokes the existence of a magnetic impurity in the QPC at low densities. Here we report on our extensive density functional calculations that reveal the formation of an electronic state with a spin-1/2 magnetic moment in the channel as the density increases above pinch-off, under very general conditions.