Grammar question - defining something additionaly

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Let's say that I have a function that is defined on some intervals, and on some it's not. I'd like to say that the interval which was not defined, was defined additionaly (because others were already defined). Should I use some prefix (sthdefined) for that or what should I use?

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You extend the function so that is also defined on the new interval. This gives you formally a new function, since functions with different domains are different.