# Condition number

## Contents

## Definition for a function of one variable

### For an arbitrary function of one variable

The condition number for a function at a point in the interior of its domain, and where is a continuous function, can be defined formally as:

where denotes the absolute value.

### For a differentiable function of one variable

Consider a function of one variable. The condition number of is defined as the absolute value of the relative logarithmic derivative of . Explicitly, the condition number of at a point in the domain of satisfying the conditions that the derivative exists, , and , simplifies to:

In cases where is continuous at and around , we may be able to compute the *limit* of this expression to obtain that condition number in cases where . Explicitly:

### For a function with one-sided derivatives

For a function that is not differentiable but has one-sided derivatives and at a point , the condition number can be defined as:

## Some example functions

Function (in terms of input variable ) | derivative | relative logarithmic derivative | condition number (itself a function of ) | limiting value as |
---|---|---|---|---|

power function for some real number (domain conditions apply) | (note that the condition number is in this case a constant function) |
|||

exponential function | ||||

logarithm function () | 0 | |||

sine function | undefined, fluctuates wildly |