Condition number

Definition for a function of one variable

For an arbitrary function of one variable

The condition number for a function $f$ at a point $x_0$ in the interior of its domain, and where $f$ is a continuous function, can be defined formally as:

$lim{sup}_{x\to x_0}\left|\frac{\frac{f(x)-f(x_0)}{f(x_0)}}{\frac{x-x_0}{x_0}}\right|$

where $|\cdot |$ denotes the absolute value.

For a differentiable function of one variable

Consider a function $f$ of one variable. The condition number of $f$ is defined as the absolute value of the relative logarithmic derivative of $f$. Explicitly, the condition number of $f$ at a point $x_0$ in the domain of $f$ satisfying the conditions that the derivative $f^{\prime }(x_0)$ exists, $x_0\ne 0$, and $f(x_0)\ne 0$, simplifies to:

$\left|\frac{x_0{f}^{\prime }(x_0)}{f(x_0)}\right|$

For a function with one-sided derivatives

For a function that is not differentiable but has one-sided derivatives $f{\text{'}}_{-}(x_0)$ and $f{\text{'}}_{+}(x_0)$ at a point $x_0$, the condition number can be defined as:

$\left|\frac{x_{0}max\{|f{\text{'}}_{-}(x_0)|,|f{\text{'}}_{+}(x_0)|\}}{f(x_0)}\right|$