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Is there any standards for when math is supposed to be upright?

E.g.

The f in f(x) is in italics but cos in \cos x is upright. As I see it, both are functions?

\int f(x) \, dx OR \int f(x) \, mathrm{d}x?

P(Y=y) OR \text{P}(Y=y) OR p(Y=y) OR \text{p}(Y=y) for probabilities?

10

Look at the guidelines for mathematical journals for advice. For example, the AMS guidelines and APS guidelines.

In general, anything considered an abbreviation (sin for sine etc.) should be upright. In the case of differentials, it matters what kind a lot.

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