Hello there,
The question can have two different answers for two different applications, but in general time would appear discrete because we can't measure time down to an infinitesimally small interval, as it is always broken up into pieces. There's no timepiece that can measure non discrete units of time (a motor clock would seem to be continuous, but then we see the magnetic field changing polarity from plus to minus which is still discrete). We can go down into the nanoseconds, picoseconds, etc., but never down to the point where there is no jump to the next interval. The catch however is in the application, where we might go down low enough to consider it continuous, but if we dont go down low enough we might consider it to be discrete.
Consider a voltage meter, analog movement. In classical mechanics the needle has the ability to be positioned at any place on the dial, and that means that we can not find an interval that it is forced to move to after it has reached a different place, but can move between any two intervals we can find on the face. So if we divide down to 0.1 inches, the needle can move between 0.2 and 0.3 inches for example, and if we choose 0.02 and 0.03 inches, the needle can still move between those two places. And if we choose 0.000002 and 0.000003 inches the needle can still rest between those two, and so on and so forth
We can never find two places that are different where the needle can not rest in between. And note that we are making a measurement here.
Quantum mechanics would differ a little, but you'd have to go to great lengths to show that there are still discrete steps because the data becomes random to some extent and so we end up with a needle placement that depends on a changing quantity where the average gives us our reading, so this would probably be ignored in all but quantum mechanics itself.
Now back to the timepiece. The gears turn in jumps as the special gear limits the movement to discrete increments (unfortunately i cant remember the name of this special mechanism in mechanical clocks). In the atomic clock the oscillations of a particular kind of atom are counted, so it's a count after all.
I guess we could get radical here and use a volt meter to measure time, using a capacitor that charges up to generate the voltage. But that's more of a cheat than anything else as it has very limited scope so we would only be measuring over a discrete time period anyway.
The more application specific view is noted by their own use of the word continuous when they describe grade point average as being continuous. I've never seen one like 3.2876475892837432324 have you