function differentiable at only one point

Let $f\colon\mathbbmss{R}\to\mathbbmss{R}$ be the function

$f(x)=\begin{cases}x,&\mbox{when $x$ is rational},\\ -x,&\mbox{when $x$ is irrational}.\end{cases}$

See this entry (http://planetmath.org/FunctionContinuousAtOnlyOnePoint). Let $g\colon\mathbbmss{R}\to\mathbbmss{R}$ be the function

$g(x)=f(x)x.$

Then $g$ differentiable at $0$ , but everywhere else non-differentiable.

Indeed, since

$\displaystyle g^{\prime}(0)$	$\displaystyle=$	$\displaystyle\lim_{h\to 0}\frac{f(h)h-f(0)0}{h}$
	$\displaystyle=$	$\displaystyle\lim_{h\to 0}f(h)$
	$\displaystyle=$	$\displaystyle 0$

$g$ is differentiable at $0$ . If $g$ would be continuous at $x\neq 0$ , then $f(x)=g(x)/x$ would be continuous at $x$ . This result (http://planetmath.org/DifferentiableFunctionsAreContinuous) implies that $g$ is non-differentiable away from the origin.

Title	function differentiable at only one point
Canonical name	FunctionDifferentiableAtOnlyOnePoint
Date of creation	2013-03-22 15:48:16
Last modified on	2013-03-22 15:48:16
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	6
Author	matte (1858)
Entry type	Example
Classification	msc 57R35
Classification	msc 26A24
Related topic	FunctionContinuousAtOnlyOnePoint