A function transforming the complex plane into the complex plane is called density continuous iff it is a continuous when the density topology is used in the domain and the range. A term strong density continuous function is self-explaining. The main result of the paper states that the complex analytic functions are density continuous. However, only functions of the form \(f(z)= a+ b z\), where \(b\) is either real or imaginary, are strongly density continuous among complex analytic functions.