Dirac delta's definition says it is zero except at $x=0$, and its Riemann integral over the real line equals $1$. Why do we define and use things in mathematics than contain contradictions in its definition?
These two properties in its definition contradict each other, because the integral of a function that has only one argument where its value isn't zero cannot be positive. You might say, ok, but mathematicians know it very well and that's why Dirac delta is not a function. It's a distribution. Fine, but is it really sufficient to define something that IS NOT a function, call it 'distribution' and use it as if it were a function? How do we know the reuslts we get are reliable if the tool we chose contains a logical contradiciton? Why can we even expect correct results?
Maybe I don't know much about distributions, but just giving Dirac delta a label 'distribution' doesn't solve the problem connected with using it as if it was a usual function.
For example, I could define a function $f(x)=0$ and say $\int_{-\infty}^{\infty}f(x)dx=1$, and then use it in my calculations, but should I rely on the results I get? Obviously not. Defining a notion is not enough and we should first see if it's correct.
Could anyone help me understand why is it perfectly okay to use Dirac delta, and show that it's different from the example I gave?