Consider a system of the form: dx/dt = f(x,y) , dy/dt=g(x,y), with the property that the associated ODE dy/dx = g(x,y)/f(x,y) has a unique solution to IVP y(0)=0.

Also, f(x,y) is smooth every *except* the point (0,0), at which it has an infinite discontinuity, and g(x,y) is continuous everywhere. Does it follow that there is a solution to the system which tends to (0,0)?

This problem may be not well-formulated, but it seems like there may be a topological argument for the existence of such a solution to the system.