A function \(f\:{\mathbb R}\to {\mathbb R}\) is a strong (internally strong) Światkowski function, if for all \(a<b\) and each \(r\) in the open interval with endpoints \(f(a)\) and \(f(b)\) there is a point \(z\in (a,b)\) such that \(z\in C(f)\) \(\bigl (z\in \int C(f)\bigr)\) and \(f(z)=r\), where \(C(f)\) is the set of all continuity points of \(f\). A function \(f\) is internally quasi-continuous if it is quasi-continuous and the set of all discontinuity points is nowhere dense. In the paper, maximums of internally quasi-continuous, Darboux internally quasi-continuous and internally strong Światkowski functions are characterized. Lattices generated by these functions are characterized, too.