Please, could you tell me what "continuum" means in this context and what are ways to rephrase it: "His motives for volunteering lie somewhere on the continuum between charitable and self-serving."

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It means scale, so if you imagine a line with one end labelled "charitable" and the other end labelled "self-serving", his motives would be somewhere between the two ends.
Other words would be "scale" or "spectrum", but you could also just say "lie somewhere between charitable and self-serving".