I might be in a grey area here, but just for the interest of everyone, how about these." It is a mathematical invention which is useful (?) in mathematics but no use for anything else."
So, you want to be a mathematician - all the best.
I did quite a lot of maths in my chem eng course.
Give me one example where you can use the word (or related adjective etc) in everyday life where it is not just an exaggeration such as "that was infinitely good".
Many adjustments are infinitely variable, volume, temperature, pressure, tuning radios etc.
Many gadgets and things are infinitely adjustable; reclining car seats are described as infinity adjustable, anything with a screw adjustment is infinitely adjustable such as a lathe, a focusing mechanism etc.
Some musical instruments have infinitely adjustable tones or notes.
Measuring instruments – thermometers, altimeter, barometer pressure sensor, strain gauge have infinitely smooth progressive readouts.
A ball thrown in the air will stay at its maximum height for an infinitely small time before falling, same with a piston (it has to stop before reversing).
The centre line of a smoothly moving clock hand will remain on the centre line of the hour mark for an infinitely small amount of time.