  In this paper, we show that a geometrical condition on $2\times2$ 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.
