Did you mean Dirac delta instead of Kronecker delta?

Yes, my mistake. Kronecker is used often in quantum mechanics, but when working in Hilbert space rather than real space.

> Under who's definition

W. Rudin, Principles of Mathematical Analysis.

W. Fleming, Functions of Several Variables.

For the set of real numbers R, a function

f: R --> R

x in R, d in R, function f is differentiable at x with value d provided for h in R

lim_{h --> 0} (f(x + h) - f(x))/h = d

In this case we write

f'(x) = df(x)/dx = d

In that case, in particular,

lim_{h --> 0} (f(x + h) - f(x)) = 0

so that f is continuous at x.

Differentiation is important in physics, e.g., in Newton's second law.

The Dirac delta function is not a function but a distribution. This is just the product of sloppy notation by the physicists.

No, it is just that physicists means something different than mathematicians when they use the word "function". Just assume that they always are distributions that are evaluated with dirac deltas and everything makes sense.

> If f is differentiable at a point x0, then f must also be continuous at x0.

https://en.wikipedia.org/wiki/Differentiable_function#Differ...