Without a doubt, you should avoid writing “without a doubt” when writing about any subject concerning which you are not 100% certain and/or about which someone else might be less than certain. ;-)
You could reasonably write “without a doubt, two plus two equals four” since no one doubts that.
It might be unwise to write, “without a doubt, no one could play any faster than that,” or “without a doubt, the tempo was allegro” – unless you checked the score to see if was marked that way.
When in doubt, don’t write “without a doubt” at all in your papers. It really does not add anything to what you write, and it looks bad if your statement turns out to be, uh, open to doubt… or just plain wrong.
Remember my motto about how specific to be in your papers: “Tell me what you do know. Do not tell me what you don’t know.”
Dan Mitchell 