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Subsection 6.1.1 Properties of Delta functions
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If \(f(x)\) is continuous in a certain interval which includes the origin, then
\begin{equation*}
\int\limits_{-\infty}^{\infty}f(x)\delta(x)\,dx = f(0)
\end{equation*}
\begin{equation*}
\int\limits_{-\infty}^{\infty}f(x)\delta(x-a)\,dx = f(a)
\end{equation*}
\begin{equation*}
\int\limits_{-\infty}^{\infty}f(x)\delta(a-x)\,dx = f(a)
\end{equation*}
\begin{equation*}
\delta(-x)=\delta(x)
\end{equation*}
\begin{equation*}
\delta(ax) = \frac{1}{a}\delta(x), \quad a \gt 0
\end{equation*}
