Reference: Continuity of Eigenvectors

The "standard" reference for all results of such types is

Kato, T., Perturbation theory for linear operators. Springer-Verlag, 1976

The finite dimensional theory is well exposed in Chapter 2, with the non-analytic dependence in section 5.

As a side remark: your statement is actually not, as your title suggests, about "continuity of eigenvectors", which is false (see Chapter 2, section 5.3 of the above citation). What you are describing is the continuity of eigenprojections.

If you want to be historical, these types of problems were originally treated by Rellich in the 1930s. A more detailed account was provided in his 1954 lecture notes which are freely available.