Cutting cylinder form a space

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We have a 3 dimensional space.
At all points of this space a value(density for example) is defined.
This value changes smoothly.

The question is: "Can I always cut a cylinder from this space, so that it's center of mass is on it's main axis?"

By "smoothly" there are 2 cases to consider:

  1. along any curve, the value "do not jump"
  2. first condition holds, moreover the derivative of the value "do not jump" too.