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When someone asks two questions and uses “or” in between, there are two options: Put a comma/semicolon before the “or” that separates the two sentences and a question mark at the end; or make it into two questions. When the questions are lengthy or there is a lot of other punctuation, it is probably best to break them into two separate sentences.
…Was it on Monday, or did you do that later in the week?
…Was that on Monday, September 5, 2017? Or did you and your husband, Bill, do that on Sunday, the 11th?
…Did you notice whether the man you were talking to was wearing a heavy jacket? Or did you not pay any attention to the clothing he had on that afternoon?