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Home < 70 Calculus Starter Lessons < 13 Lessons on Limits and Continuity << 5 Jumps and absence of unlimited error control
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Jumps and Limited Error ControlIn 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 conditionThis means as the input x to the function y = f(x) becomes a better approximation to the number a, there is no guarantee the difference |f(x)-f(a) | will be smaller than the error control target e. This concept is illustrated by functions whose graphs have a few jumps in them. The height of the largest jump near a point x = a indicates how small the target tolerance e or ½ ·10-n can be in the discussion of error control.
Unlimited error control is possible in the following circumstances: These circumstances appear when f(x) is continuous at x = a. 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 \[ |x-a|\lt \frac12\cdot\frac1{10^m} \quad\mbox{implies} \quad |f(x)-f(a)|\lt \frac12\cdot\frac1{10^n} \] governing error cannot be guaranteed for n > 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.
On a computing machine which computes to at most n0 decimal places, the error control of a single addition and multiplication are guaranteed to only n0 binary (or decimal) places. Digits beyond the n0 place are uncertain. If several such calculations are done, with numbers in one calculation being used in the next, errors accumulate and accuracy is lost. The calculations in question may have to be reorganized to improve accuracy. |
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Home < 70 Calculus Starter Lessons < 13 Lessons on Limits and Continuity << 5 Jumps and absence of unlimited error control
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