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I am currently studying mathematics at College A (In the United States). Assume that College A is not a notoriously "prestigious" college, but that it's also not a "nobody" college. When applying to graduate schools for mathematics, is College A a safety school just because I went there for undergraduate?
I think it's important to assume that the college isn't some top 20 school because I think that would not be the case.
I was surprised to find no information on this topic online.
If by "safety school" you mean that you'll almost surely be accepted to the grad program... I think the answer is "no". Even if you've done reasonably well as an undergrad, for grad admissions you are suddenly competing with people from all around the world. The "playing field" is suddenly much different.
Yes, some universities do try to take good care of their own grads, but this is very inconsistent from place to place, and depends enormously on the current administrative cadre at a given place.
Also, typically R1 universities graduate many more undergrad math majors each year than they have openings for (funded!!!) grad students. So the numbers/money game alone restricts what your home place can do.
So, in summary, if you are one of the better math majors in your year, it would be sensible to apply your own place as a back-up. But/and you might want to make it clear to your own dept what you are doing, since many depts will simply not make an offer to prospective students who give every indication that they wish to go elsewhere... since the current rules/guidelines for grad admissions in the U.S. effectively punish programs for making offers to students who won't accept.
Talk to your faculty at your place about the advisability of this...
Talk to a few current math professors to get an opinion on whether you would be a good candidate for their graduate program. If they say yes, more or less unanimously, then, yes, you could use it as a sort of backup in case other applications fail to get results.
But, in some sense, it doesn't really matter. No one can guarantee you admission anywhere.
And don't get complacent. Make a solid application there as well as elsewhere.
There is an advantage to going to a new place, actually, as you get the opportunity to meet new people (faculty) and with it, perhaps, new ideas. I always advised my students not to continue at my institution for that reason.
