Try this:

Improvise, but have it so that at no point are you not fretting a note.

It could be any finger, but there must always be contact with the string.

Next, try it so that there is always some space between every note.

It can be any length of space, but it must always be there.

Why?

It’s an exercise designed to increase your awareness of note length.

Specifically, I use it when I feel a student is unaware that they might be, for example, not playing any long notes, or just letting notes die off.

You could apply it to any of the things that we all do but don’t pay enough attention to – vibrato, for instance.

Apply the exercise to that: Play, but vibrate all the notes, or play, but use no vibrato.

It’s all about control.