This paper considers the Cauchy problem for two-dimensional compressible Euler equations, written in the following form: $$ \cases \rho_t +{\rm div}\big((\rho_0 + \rho)u\big) = 0,\\ u_t + u\cdot\nabla u + \nabla h +\frac{h_0+h}{\rho_0+\rho}\nabla\rho =0,\\ h_t + u\cdot \nabla h + (\gamma-1)\big(h_0+h\big){\rm div}\, u =0. \endcases \tag1 $$ Here, $\gamma>1$, $\rho_0>0$, $h_0>0$ are constants, and (1) is augmented by the initial conditions: $$ \big(\rho, u, h\big)(t, \cdot) = \big(\overline\rho, \overline u, \overline h\big)\in H^s(\Bbb{R}^2, \Bbb{R}^4). \tag2 $$ \par The main result states that for every $s>2$ the data-to-solution map $$ H^s(\Bbb{R}^2, \Bbb{R}^4)\ni \big(\overline\rho, \overline u, \overline h\big) \mapsto \big(\rho, u, h\big)\in {C}\big([-T,T], H^s(\Bbb{R}^2, \Bbb{R}^4)\big) $$ for (1)--(2) is not locally uniformly continuous.