The Mobius Strip is Not a Smooth Surface

The Mobius strip is not smooth because the normal vector on the surface does not vary continuously as we move it along the surface, so if it were to move around once, it would move back onto itself.. To see this take a Mobius strip and draw on a normal at any point P. Move the normal along the strip until it reaches the point P again.

The normal will now point in the opposite direction as before. The Mobious strip is continuous on each part of its surface, but not on the whole surface.