Understanding the Dirac Delta function

[redtree]

I just want to make sure that I am understanding the Dirac Delta function properly. Is the following correct?:

For two variables $x$ and $y$:
\begin{equation}
\begin{split}
\delta(x-y) f(x) &= f(y)
\end{split}
\end{equation}

And:
\begin{equation}
\begin{split}
\delta(x-x) f(x) &= f(x)
\end{split}
\end{equation}

 

[Vanadium 50]

Neither of those is correct.

 

[Ssnow]

The Dirac delta is not a function in the traditional sense, it can be rigorously defined either as a distribution or as a measure.
Ssnow

 

[FactChecker]

The Dirac delta function is defined as a generalized function by its behavior in an integral: ∫f(x)δ(x)dx = f(0).
So in your question, ∫δ(x-y)f(x)dx = f(y) would make better sense than your expression (1). The integral can not be omitted. Your expression (2) does not make sense to me.