Why is dirac delta function 0 except at 0

Question

The dirac delta function's definition is following:definition of dirac delta function

But why dirac delta function is 0 except at 0?? How can be this property derived

Answer 1

A distribution $T$ is said to vanish (or have value $0$) on a region $U$ if $T\{f\}=0$ for every test function $f$ with support on $U.$

Since this is the case for $T=\delta$ and $U=\mathbb{R}\setminus\{0\}$ the Dirac delta distribution $\delta(x)$ is considered to vanish for $x\neq 0.$

Answer 2

In many physics textbooks they tend to define the Dirac-Delta distribution as the piecewise function $$\delta(x) = \begin{cases}0, \ \ \ &x\neq 0\\ \infty, \ \ \ &x=0. \end{cases}$$ This is how one could possibly define this distribution in some sense.