This may be just a grammatical question.
I have seen the following two representations:
- This function is continuous in $D.$
- This function is continuous on $D.$
Is there any difference between them? I thought that "in $D$" means the inside of $D$ while "on $D$" contains its boundary, but I have faced many exceptions.
Could anyone explain the difference? I would appreciate if you could tell me other cases where this kind of difference matters.