It is always a fundamental physical quantity.

Time is a physical quantity. It's so fundamental that we need it to make changes comparable. 

For time it is not important that it is continuous or not.  Time is graphically best showable as straight line with signed begin- end end-point. This line can't be interrupted because time isn't stopping or beginning.  Only a measurement of time for a predefined process has a defined beginning and ending.

Physical time is always constant - time per se can't be faster or slowlier!

Mental time feeling of Human Being is quite different - of course. 

Time may be a diskret moment - this fact is important if you want compare f. i. Day Time worldwide. 

Time is no physical property of any physical object. Time - measured worldwide unified

by worldwide unified clocks and units is a physical attribute of every change in nature.

In my view 'time' is perceived as a physical quantity attributed to change in distribution of matter or state of the matter through a physical/chemical phenomena happening in the space. We have well known term "Space quantization" and also have upper limit on velocity of any event then time must be discrete because it is the span in which any event or the information travel form one point to another point in the space. 

       In my view if the observer is also moving/ changing (more appropriately evolving) in the same direction in the event space with same rate at which event is moving then observer will not perceive any change and for that particular observer time may be constant.

       If there is mismatch between event clock and observer clock then only observer is able to attribute this change to time.