Dirac Delta Function

The dirac delta function is defined as follows:

$\delta (t)={\begin{cases}\infty &t=0\\0&{\text{otherwise}}\end{cases}}$

Properties

The Dirac Delta Function satisfies the following properties:

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(t-c) = 0 \,\!} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \ne c}
2. $\int _{c-\epsilon }^{c+\epsilon }\delta (t-c)\,\mathrm {d} t=1$ for any $\epsilon >0$
3. $\int _{c-\epsilon }^{c+\epsilon }f(t)\,\delta (t-c)\,\mathrm {d} t=f(t)$ for any $\epsilon >0$