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We consider the Cauchy problem for the inhomogeneous nonlinearSchr\"{o}dinger (INLS) equation \[iu_{t} +\Delta u=|x|^{-b} f(u),\;u(0)\inH^{s} (\mathbb R^{n} ),\] where $n\in \mathbb N$, $0<s<\min \{ n,\; 1+n/2\} $,$0<b<\min \{ 2,\;n-s,\;1+\frac{n-2s}{2} \} $ and $f(u)$ is a nonlinear functionthat behaves like $\lambda \left|u\right|^{\sigma } u$ with $\sigma>0$ and$\lambda \in \mathbb C$. Recently, An--Kim \cite{AK21} proved the localexistence of solutions in $H^{s}(\mathbb R^{n} )$ with $0\le s<\min \{ n,\;1+n/2\}$. However even though the solution is constructed by a fixed pointtechnique, continuous dependence in the standard sense in $H^{s}(\mathbb R^{n})$ with $0< s<\min \{ n,\; 1+n/2\}$ doesn't follow from the contraction mappingargument. In this paper, we show that the solution depends continuously on theinitial data in the standard sense in $H^{s}(\mathbb R^{n} )$, i.e. in thesense that the local solution flow is continuous $H^{s}(\mathbb R^{n} )\toH^{s}(\mathbb R^{n} )$, if $\sigma $ satisfies certain assumptions.

Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schr\"{o}dinger equation in $H^{s} (\mathbb R^{n} )$