Summary: Let \(\gamma\) be a non-rectifiable closed Jordan curve in \(\mathbb{C}\), which is merely assumed to be \(d\)-summable (\(1<d<2\)) in the sense of Harrison and Norton. We are interested in the so-called jump problem over \(\gamma\), which is that of finding an analytic function in \(\mathbb{C}\) having a prescribed jump across the curve. The goal of this note is to show that the sufficient solvability condition of the jump problem given by \(\nu > \frac{d}{2}\), being the jump function defined in \(\gamma\) and satisfying a Hölder condition with exponent \(\nu\), \( 0<\nu\leq 1\), cannot be weakened on the whole class of \(d\)-summable curves.