Dirac delta

The Dirac delta function $\delta (x)$ satisfies the following property:

$\int _{-\infty }^{\infty }f(x)\delta (x-y)dx=f(y)$

for all functions $f(x)$. This can be seen as the continuous analogue of Kronecker Delta. Taking the function $f(x)=1$ for all $x$, using the above property and letting $y=0$:

$\int _{-\infty }^{\infty }\delta (x)dx=1$

Therefore, the Dirac delta function is normalized.

Technically speaking, no function satisfies the first property above. However, the notion of Dirac delta can be made mathematically rigorous. In practice, it can be seen as the limit of a function which becomes extremely concentrated at a single point. Thus it is often said that:

$\delta (x)={\begin{cases}\infty &{\mbox{if }}x=0\\0&{\mbox{if }}x\neq 0\end{cases}}$

For $\mathbb {R} ^{n}$ the Dirac delta function can be generalized to:

$\int _{\mathbb {R} ^{n}}f(\mathbf {x} )\delta (\mathbf {x} -\mathbf {y} )d^{n}\mathbf {x} =f(\mathbf {y} )$

References

Weisstein, Eric W. "Delta Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DeltaFunction.html