Jumps and Limited Error Control

In some cases unlimited error control is not possible at the point x = a. It fails in the following case:

There is an e > 0 such that for every d > 0, there is some x satisfying the condition 

|x-a| < d and |f(x)-f(a)| > e.

Again, unlimited error control is possible in the following circumstances:

For each target tolerance e > 0, there is a tolerance d > 0 such that the condition 

|x-a| < d and |f(x)-f(a)| £  e.

Computations on machines with finite accuracy precision arithmetic, restrict the number n of decimals places that can be accurately computed. Every computing machine which calculates to finitely many binary or decimal places, suffers from such a limit. Small discontinuities in calculations appear, except in those case where exact arithmetic can be done. For example, on a computing machine which computes to at most n₀ decimal places, the existence of a rule of the form

|x-a| < 1 2 1 10m     implies    |f(x)-f(a)| < 1 2 1 10n

governing error cannot be guaranteed for n ³ n₀ and can be considered improbable for most functions evaluated numerical by a computer. An exception is provided by functions whose numerically values can be represented (or encoded) exactly on a machine.