I'm assuming that by continuous movement, you mean a constant rate of rotation. In that case:

The position of the body over time can be expressed as <r*cos(t*s), r*sin(t*s)>, where s is the angular speed (magnitude of angular change per time) and r is the distance from the center of rotation. The velocity of that vector is its derivative, <r*s*-sin(t*s), r*s*cos(t*s)>. This is not a constant expression, since its value still varies as t changes. So the strict answer to your question, Austin, is no, the velocity of an object following a circular trajectory is not constant. However, there is a difference between velocity and speed - while velocity is an expression of where an object is going at any given time, 'speed' is a measure of how quickly it's getting there. Speed is given as the magnitude of velocity; graphically, the length of a body's velocity vector. It can be shown that the magnitude of the velocity mentioned above is simply r*s, a constant value. So an object travelling in a circle at constant angular velocity has a constant speed. Also note that that speed is a product of both the angular velocity and the radius of the circle - the bigger the circle is, the faster the edge goes when you spin it.

Hopefully I've been able to answer the question to your satisfaction..how'd I do?