Interpreting ##\alpha \delta(x)##: Is ##\alpha## at ##x=0##?

[matematikuvol]

In Dirac definition $\delta(x)$ is $\infty$ when $x=0$, and $0$ when $x\neq 0$. My question is when I have some $\alpha \delta(x)$ could I interpret this like function which have value $\alpha$ in point $x=0$?

 

[Fredrik]

That interpretation would suggest that
$$\int \alpha\delta(x)=0$$ which is wrong. The right-hand side should be 1. Makes more sense to think of that value as $\alpha\cdot(+\infty)=+\infty$, if $\alpha>0$. Note that since
$$\int\delta(x)f(x)dx=f(0)$$ for all f, we should have
$$\int \alpha\delta(x)f(x)dx=\int\delta(x)(\alpha f)(x)dx =(\alpha f)(0)=\alpha f(0).$$