# 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 FunctionDifferentiableAtOnlyOnePoint 2013-03-22 15:48:16 2013-03-22 15:48:16 matte (1858) matte (1858) 6 matte (1858) Example msc 57R35 msc 26A24 FunctionContinuousAtOnlyOnePoint