Model continuity plays an important role in applications like system identification, adaptive control, and machine learning. This paper provides sufficient conditions under which input-output systems represented by locally convergent Chen-Fliess series are jointly continuous with respect to their generating series and as operators mapping a ball in an $L_p$-space to a ball in an $L_q$-space, where $p$ and $q$ are conjugate exponents... The starting point is to introduce a class of topological vector spaces known as Silva spaces to frame the problem and then to employ the concept of a direct limit to describe convergence. The proof of the main continuity result combines elements of proofs for other forms of continuity appearing in the literature to produce the desired conclusion. (read more)