Let $H$ be a separable Hilbert space. We write $\sigma (A)$ for the spectrum of $A\in B(H)$, $\sigma_a(A)$ and $\sigma_{ea}(A)$ for the approximate point and the essential approximate point spectrum of $A$. Operator $A\in B(H)$ is quasihyponormal if $\| A^*Ax\| \le \| A^2x\|$ for all $x\in H$. In this paper we show that the approximate point spectrum $\sigma_a$ and the essential approximate point spectrum $\sigma_{ea}$ are continuous in the set of all quasihyponormal operators.