When is the sum of E[x] equal to E[sum(x)]?

Question

This equivalence was used in my statistics course to prove that the sample mean is an unbiased estimator when the sample has:

1. a mean "mu"
2. a variance "sigma^2"

Proof in my book

On Wikipedia I found that this equality doesn't always hold true:

https://en.wikipedia.org/wiki/Expected_value

(Under Uses and appliances it says that: https://wikimedia.org/api/rest_v1/media/math/render/svg/b866cf6a1cae9adf1d0fee60b0c0fe0633c6b9f1)

Is there a general rule for when E[g(x)] = g(E[x]) ?