# function differentiable at only one point

Let $f:\mathbb{R}\to \mathbb{R}$ be the function

$$f(x)=\{\begin{array}{cc}x,\hfill & \text{when}x\text{is rational},\hfill \\ -x,\hfill & \text{when}x\text{is irrational}.\hfill \end{array}$$ |

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

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

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

Indeed, since

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

$=$ | $\underset{h\to 0}{lim}f(h)$ | |||

$=$ | $0$ |

$g$ is differentiable at $0$. If $g$ would be continuous at $x\ne 0$, 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 |