Velocity/Talk

Question: If a particle has a defined position at every time, must it necessarily also have a defined velocity? Consider a particle moving along a line, so its position along the line at time t is x(t). Suppose we define x(t) as follows:

          { 1   if t > 0
  x(t) =  {
          { 0   if t <= 0

If I remember my calculus correctly, x'(0) is undefined, while x(t) = 0. Does it therefore follow that at time t=0, the particle has a position and is moving but has no velocity? Would it be physically possible (i.e. compatible with the laws of physics as we currently understand them) for a particle with that behaviour to actually exist? -- SJK