What is the Effect of Multiplying a Function by the Unit Impulse Function in the Frequency Domain?
I think there is a slight typo in robert bristow-johnson's answer. Should be

\begin{align}
x(t) &amp;= \frac{1}{2 \pi} \int\limits_{-\infty}^{\infty} X(\omega) e^{j \omega t} \, \mathrm{d}\omega\\
\\
 &amp;= \frac{1}{2 \pi} \int\limits_{-\infty}^{\infty} H(\omega_0) \delta(\omega - \omega_0) e^{j \omega t} \, \mathrm{d}\omega\\
\\
 &amp;= \tfrac{1}{2 \pi} H(\omega_0) e^{j \omega_0 t}
\end{align}

What this actually means is that during the inverse transform, $\delta(\omega - \omega_0)$ selects point frequency $\omega_0$ from the frequency response $H(\omega_0)$, and the value of $H(\omega_0)$ at that point determines the magnitude and phase of the time domain sinusoid $e^{j \omega_0 t}$.