We prove that for the functions of the form \(g(x) = h(x) + \frac{C}{x+i}\) , where h belongs to the continuous Wiener algebra \(W_0\) , the intersection of the Gabor frame set \(\mathcal {F}_g\) with every hyperbola \(\{\alpha , \beta > 0 \mid \alpha \beta = c\}\) is open in the relative topology. In particular, this applies to all rational functions g. We also show that the same phenomenon occurs for the functions from the Wiener algebra W which are discontinuous at one point.
