LLT71 said:
The argument given there is not a proof because it assumes at the outset that the integrand ##f(x) \delta (x -t) ## is a function that has an integral. If we use the definition of "integral" given in introductory calculus ( e.g. as defined terms of the limit of Riemann sums) together with the definition that ##\delta(x-t) = 0## when ##x \ne t##, we can prove ##\int_{-\infty}^{\infty} f(x)\delta(x-t) dx = 0##. That contradicts what we want to be true.
The fact that the dirac delta "function" is commonly defined with the condition ##\int_{0 -a}^{0+a} \delta(x) dx = 1 ## shows the delta function is not actually a function. For, example, what would you say to a calculus student who defined a function by saying: " I define ##f(x)## as ##f(x) = x^2## and ##\int_{-1}^{1} f(x) = 13.7 \ ##"?
Once you define a function, you don't have any choice about what its integrals are going to be - except that you can choose different definitions of the general concept of integration.
To approach the dirac delta function coherently, we must revise the definition of integration - or at least the notation for integration.
One way to do this is to define the notation ##\int_{a}^{b} f(x) \delta(x) dx ## to mean something besides the usual kind of integral. This has the drawback of making the notation "##\int##" ambiguous, but that is a frequent transgression in writing mathematics.
Let ##\{g_n\}## be the sequence of functions given by ##g(n) = \frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{x^2}{2 \sigma^2}} ## where ##\sigma_n = 1/n##.
(This defines ##g_n## to be the probability density for a normal distribution with mean zero and standard deviation ##\frac{1}{n}##. Like any probability density, ##g_n## has the property that ##\int_{-\infty}^{\infty} g_n(x) dx = 1 ## .)
The graph of these functions is illustrated by the curves that
@micromass showed. As ##n## increases they become more and more peaked over the value ##x = 0##.
Then we
define the notation ##\int_{a}^{b} f(x) \delta(x) dx ## to mean ##\lim_{n \rightarrow \infty} \big{[} \int_{a}^{b} f(x) g_n(x) dx \big{]} ## if that limit exists.
Notice that this approach defines an integral with the symbol "##\delta(x)## in the integrand without defining the symbol "##\delta(x)##" to have any meaning by itself.
There are probably more sophisticated was to define ##\delta(x)## than the method I illustrated, but the method I illustrated indicates the sort of thing you must do to have a logically consistent structure.
