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This paper deals with connections on $p$-adic analytic curves, in the senseof Berkovich. The curves must be compact but the connections are allowed tohave a finite number of meromorphic singularities on them. For any choice of asemistable formal model of the curve, we define an intrinsic notion ofnormalized radius of convergence as a function on the curve, with values in$(0,1]$. For a sufficiently refined choice of the semistable model, we provecontinuity and logarithmic concavity of that function. We characterize\emph{Robba connections}, that is connections whose sheaf of solutions isconstant on any open disk contained in the curve.

Continuity of the radius of convergence of differential equations on $p$-adic analytic curves