It is known that the Dirac delta function scales as follows:$$\delta(kx)=\frac{1}{|k|}\delta(x)$$ I have studied the proof for it, considering Dirac delta function as a limit of the sequence of zero-centred normal distributions (as given here).
However, when intuitively thought about it, this does not seem correct. Since $\delta(x)$ is zero everywhere except at $x=0$, $\delta(kx)$ should also be zero for any non-zero value of $x$ (given $k\in R-\{0\}$). Also for $x=0, kx=0$, and, thus, $\delta(kx)=\delta(x)$.
From the above logic it is evident that the scaling property should be the following.$$\delta(kx)=\delta(x)\forall x\in R, k\neq 0$$ However, as we know this is not true, can you point out where I am going wrong in thinking like this. Please note that I do not require some other kind of proof (until necessary), just a flaw in this kind of thinking.