What is the rate at which a single note would have to be played, such as on a guitar, in order to sound continuous?

Just as you can make a motion picture from a series of still images, you can make the illusion of continuous sound if a single tone is repeated quickly enough. In music it is called tremolo and its speed on a guitar is limited by the human hand. Electronic devices can do this though. I have a synthesizer with that function. I don't have a precise readout of the exact speed but I suspect that it is well over 100 repetitions per second.

Russell E is spot-on. With an electric rig, a hot amp can do it, vis Jimi Hendrix. On an acoustic, the best, in the hands of a very skilled player, can sustain a note nicely, but still not a long time. Since you should give Russ a BA, here's a different approach, with totally insane note reps...

https://www.youtube.com/watch?v=-y3h9p_c5-M

Some troll moved you to Mathematics. I'm moving you to Performing Arts.

That's kind of a tricky question. With enough sustain you could pick pretty slowly on electric guitar and still never have the note die. But you would also hear the attack of each pick, so while there would be constantly sound, it would fluctuate in dynamics. No one could tremolo pick fast enough that you couldn't hear the attack of each note.

That is what is known as sustain. There is no finger or picking technique that will make more than one note sound continuous.

And if playing an acoustic, the note will decay very rapidly.