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baby oil, to my knowledge, is just mineral oil with a bit of fragrance added. You should be fine since I don't think that they would add a fragrance that would make a baby's skin irritated, so...

I use mineral oil daily, no matter how often I use my blade. I've calculated recently that it costs me $0.007/day to keep my razors oiled. Cheapest insurance you'll ever buy.

For hard to reach...