Integration methods...an error-proof method?

[AlexFekken]

In the mean time (lazy Sunday afternoon), I have come to realise that this is not just an issue when there are "discontinuities" or other types of singularities are in play.

This is one way how you can "trick up" (as suggested by Pauli) most algorithms:

Select an arbitrarily small but postive number eps and an arbitrary "value" V. For example chose eps as the smallest positive number that your system can represent, and V as the largest...

You can then construct a function that has the following properties:

- it is arbitrarily smooth everywhere (i.e. it can be differentiated everywhere and as often as you like; this also implies continuity of the function and all of its derivatives everywhere)
- the function is 0 everywhere except inside the open interval (0, eps)
- the integral over [0, eps] is V
- you can make it unimodal (a single bump), i.e. you need no weird oscillations and/or jumps to construct it.

I have only added the unimodal bit so that you do not think that this would have to be a very special, "artificial" function... Explicit details for constructing such a function are in many textbooks on analysis; typically, only the exponential function is used beyond +, -, * and / (think exp(-1/x^2) and be creative).

Now you can add this "bump" to an arbitrary function f to obtain a new function g so that

- g = f everywhere except inside the tiny interval (0,eps)
- g is just as smooth as f
- the integral of g, over any interval including [0, eps], differs by V from the integral of f

In other words: as long as you don't investigate inside (0,eps) you cannot distinguish between f and g and yet their integrals differ by as much as you like.