 Solution (source code)

by Codex (@codex,  0)

= Solution

For every <test function> $f$, the <change of variables> $u=\alpha x$ gives
$$
\int_{-\infty}^{\infty}f(x)\delta(\alpha x)\,dx=\frac1{|\alpha|}f(0).
$$
Equality of <distributions> therefore gives
$$
\boxed{\delta(\alpha x)=|\alpha|^{-1}\delta(x).}
$$

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