What is wrong with the two difference usages of a function? If you treat function as a system that produces an output given an input - does it matter if it is a graph or a probability distribution?
Because that's wrong. For example, take the Dirac delta function (point mass). You can only integrate it, not evaluate it.
GCR was saying that it's OK to use the same notation for both kinds of function. Mathematicians use separate notations, which is ugly and awkward. Engineers and physicists conflate these objects in their terminology, which is misleading. The right answer is to use the same notation but explain there's a difference.
I'm leaning to something like: (X (R -> R) ((R -> R) -> (R -> R))) -> R where we are given a function (R -> R) on which it acts and a linear operation (R -> R) -> (R -> R), like integration, which acts on it and gives us a single real number as the answer.
You don't really, the same way that a regular function R->R has no specific type information that tells you ti's linear or not, e.g. both x and sin(x^2) have the same type.
The other obvious problem is its value at the origin. Infinity as a value is not something mathematicians like - at least not the way it is used here. And integrating it to get 1 makes no sense mathematically merely by setting its value to infinity.
For example, delta can be put as initial condition for psi function in the time-dependent Schroedinger equation to search for Green's function. Evolution of this initial condition in time is a regular oscillatory function that looks nothing like 0 or almost 0 almost everywhere for any $t>0$. So if it isn't 0 almost everywhere for t>0, why should we impose that at t=0.
No one integrates it over 0, but they do the delta function.
> I recall that in my analysis class, they wanted to define the dirac delta as the limit of a sequence of functions.
In my engineering courses, they often tried to make it a limit of the "rectangle" function. Consider a function that is 0 everywhere except between [-delta, delta], where its value is whatever is needed to make the area under this rectangle 1. Then let delta approach 0.
This definition still fails using the standard limits/analysis that is taught (convergence problems, etc). In mathematics, the delta function needs the theory of distributions to "work".