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In this talk, we show that   a geometrical condition on 2 × 2 systems of conservation laws leads to  non-uniqueness  in the class of 1D continuous functions. This demonstrates that the Liu Entropy Condition alone is insufficient to guarantee uniqueness, even within the mono-dimensional setting. 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 dimension. This is a joint work with Robin Ming Chen, and Cheng Yu.