Summary: In this paper, the right upper semicontinuity at \(p=1\) and continuity at \(p=\infty \) of the set-valued map \( p\rightarrow B_{\Omega,\mathcal{X},p}(r) \), \( p\in[1,\infty] \), are studied, where \( B_{\Omega,\mathcal{X},p}(r) \) is the closed ball of the space \( L_p(\Omega,\Sigma,\mu;\mathcal{X}) \) centered at the origin with radius \(r\), \(\Omega,\Sigma,\mu)\) is a finite and positive measure space, \( \mathcal{X}\) is a separable Banach space. It is proved that the considered set-valued map is right upper semicontinuous at \(p=1\) and continuous at \(p=\infty \). An application of the obtained results to the set of integrable outputs of the input-output system described by the Urysohn-type integral operator is discussed.