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Why do we use "non-decreasing" as a synonym for increasing? The negation is confusing. E.g., the sine function is not decreasing. Is the distinction between strictly and non-strictly increasing meaningful?

The distinction is often important, yes.

Sure, of course it can matter, but do we need totally different terms? Wouldn't it make more sense to add a qualifier (strictly increasing vs. non-strictly increasing).

I mean you can say "non-strictly increasing" instead of "non-decreasing" but a lot of people don't.