I have a functional equation which I have solved for rational numbers. Now to 'elevate' my solution to reals, I use the given fact that f is continuous. I could have done the same process, if I knew that f is monotone. What I would like to know is the list of criteria which enables me to make this 'elevation'. For example, two specific criteria are-
- If the function is continuous.
- If the function is monotone.
Also, I would like to know if it is enough to prove these criteria within a certain range. Thanks in advance.
In some cases e.g. the Cauchy functional equation, you can "elevate" with just the assumption that the function is measurable.