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We prove a continuous Freiman's $3k-4$ theorem for small sumsets in$\mathbb{R}$ by using some ideas from Ruzsa's work on measure of sumsets in$\mathbb{R}$ as well as some graphic representation of density functions ofsets. We thereby get some structural properties of $A$, $B$ and $A+B$ when$\lambda(A+B)<\lambda(A)+\lambda(B)+\min(\lambda(A),\lambda(B))$. We also givesome structural information for sets of large density with small sumset andcharacterize the extremal sets for which equality holds in the lower bounds for$\lambda(A+B)$.

Small sumsets in real line : a continuous 3k-4 theorem