I am studying "Partial Differential Equation. An introduction." by Walter Strauss and at the beginning of Ch. 12, while talking about distributions, it says the delta function "is not a function. It's a more general object called a distribution. A function is a rule that assigns numbers to numbers. A distribution is a rule (or transformation, or functional) that assigns numbers to functions."
It goes on to say: "The delta function is the rule that assigns the number f(0) to the function f(x)."
Can someone clarify that last statement?
I understand that f(x) is referred to as a "test function"; what is the definition of a test function?
Thanks.
$$\lim_{\alpha\to 0}\int_{-\infty}^{\infty}\delta(x,\alpha)f(x)dx = \int_{-\infty }^{\infty}f(x)\delta (x) dx = f(0)$$ take this as a definition to avoid calculation For example : $$ let \ \delta(x,\alpha)=\frac{1}{\alpha\sqrt{2\pi}}e^\frac{-x^{2}}{2\alpha^{2}} $$ \begin{align} \lim_{\alpha\to 0}\int_{-\infty}^{\infty}\delta(x,\alpha)\, f(x) \, dx &= \int_{-\infty }^{\infty}f(x)\delta (x) dx = f(0) \\ &=\lim_{\alpha\to 0}\int_{-\infty}^{\infty}\frac{1}{\alpha\sqrt{2\pi}}e^\frac{-x^{2}}{2\alpha^{2}}f(x)dx\\&= \lim_{\alpha\to 0}\int_{-\infty}^{\infty}\frac{1}{\sqrt{\pi}}e^{-y^{2}}f(\sqrt{2}\alpha y)dy\\&=\lim_{\alpha\to 0}f(\sqrt{2}\alpha \xi)\int_{-\infty}^{\infty}\frac{1}{\sqrt{\pi}}e^{-y^{2}}dy\\&=f(0) \end{align}