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**ThePerfectHacker** If $\displaystyle f$ is continous differenciable (on some open set) it means $\displaystyle f$ is differenciable and $\displaystyle f'$ is continous (on this open set). Write $\displaystyle f = g + hi$. Remember that $\displaystyle f ' = g_x + h_x i$. Thus, $\displaystyle g_x,h_x$ are continous on this open set. However, by Cauchy-Riemann equations $\displaystyle g_x=h_y$ and $\displaystyle h_x = -g_y$ so $\displaystyle h_y,g_y$ are continous also. Thus, the partial derivatives must be continous.