Do Most Math Graduates Have Strong Knowledge in All Fields?

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h00zah
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with regards to my topic, i mean that in a sense that the subject is not some sort of prodigy/genius, but an average human being with a strong interest in mathematics who wants to pursue a career in math. do most people who graduate with a math degree have a strong knowledge in all math fields, or do you come out being stronger in one particular math field and choose to pursue that route? when i hear the word 'mathematician' i think of someone who when asked a math question, no matter what field it relates to, can answer it fully, if not then to a certain extent. the reason i ask is because math just seems so overwhelming, like it would take my life 10 fold just to know half the stuff relating to math. also, there are some fields i really enjoy and others not so much. and how important is it to know definitions word by word?
 

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You can "derive" a lot of definitions if you understand the motivation. Well...in some courses.
Definitions are imporant. One can't even ask or understand a question without good definition.

There are a few underlying topics that sort of pop up everywhere. I imagine in your first year/s you'll end up doing a course on most branches.
Stuff like groups, toplogy, metric spaces, and some analysis. Technically pure fields, but they're behind all the applied topics too. So there's no escape (and why would you want to?)

Probably depends on the university you study at, but you're unlikely to completely specialize in an undergrad degree.