Continuity of eigenvalues and spectral radius for a general matrix

All of these are continuous, since they are the compositions of continuous functions.

The function from the matrix to any coefficient of the polynomial is itself a polynomial on the entries of the matrix, which is continuous. Thus, the function from a matrix to the vector listing the coefficients of the polynomial is continuous. So, the function from a matrix to its characteristic polynomial is continuous.

The function from a characteristic polynomial to its roots is continuous. So, by the continuity of composition, the function from a matrix to its eigenvalues is continuous.

The function $x \mapsto |x|$ is continuous, as is $(x_1,\dots,x_n) \mapsto \max\{x_1,\dots,x_n\}$. So, the function that yields the largest absolute value of an entry of a vector is continuous. So, by composition, the spectral radius function is continuous.