In this talk, we will show that a geometrical condition on \(2 \times 2\) systems of conservation laws leads to nonuniqueness in the class of 1D continuous functions. This demonstrates that the Liu Entropy Condition alone—which ensures the uniqueness of small BV solutions—is insufficient to guarantee uniqueness in the continuous setting, even within the mono-dimensional frame. We provide examples of systems where this pathology holds, even if they verify stability and uniqueness for small BV solutions. Our proof is based on the convex integration process. Notably, this result represents the first application of convex integration to construct non-unique continuous solutions in one spatial dimension. This is a joint work with Alexis Vasseur and Cheng Yu.