How can we solve a pair of equations that have to be solved self-consistently (even by plotting them simultaneously) .
y = 1/2 + 1/\[Pi] ArcTan[c (0.5 - x)]; x = 1/2 + 1/\[Pi] ArcTan[c (0.5 - y)];
c=0.5 we must have a plot as below one in which the vertical axis is
y and horizontal one is devoted to
c=Pi we must have