  In the paper, we consider the initial value problem to the Camassa-Holm
equation in the real-line case. Based on the local well-posedness result and
the lifespan, we proved that the data-to-solution map of this problem is not
uniformly continuous in nonhomogeneous Besov spaces in the sense of Hadamard.
Our obtained result improves considerably the result in \cite{H-K}.
