Are the real eigenvalues of real symmetric matrices continuous?

The set of real numbers is a subset of the set of complex numbers, if we consider that real numbers are complex numbers with imaginary part equal to zero. Therefore, whatever holds for all complex numbers holds for real numbers.

As you observe, a polynomial might not have real roots. However, all the eigenvalues of a symmetric real matrix are real. By definition, they are the roots of the characteristic polynomial, so you can be sure that an example like the one you proposed will not arise.