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We show that in contrast to the case of the operator norm topology on the set of regular operators, the Fuglede-Kadison determinant is not continuous on isomorphisms in the group von Neumann algebra $𝒩\left(\mathbb{Z}\right)$ with respect to the strong operator topology. Moreover, in the weak operator topology the determinant is not even continuous on isomorphisms given by multiplication with elements of $\mathbb{Z}\left[\mathbb{Z}\right]$. Finally, we define $T\in 𝒩\left(\mathbb{Z}\right)$ such that for each $\lambda \in ℝ$ the operator $T+\lambda ·{\mathrm{id}}_{l^2\left(\mathbb{Z}\right)}$ is a self-adjoint weak isomorphism of determinant class but...