Whilst working with the equations for the previous post on the fusc function, I encountered one I wished to invert. That equation was essentially

but I had not noticed that this equation is special, and that the ‘2’ is significant.

It turns out that the inverse is identical:

Similarly, the inverse of

is given by

but note the switch from *floor* to *ceiling* in this case.

I wondered if equations slightly more general than this had a solution, and it seems that the answer is essentially ‘no’. To be more specific, consider the equation

Following is a table of some properties of this function of *x*, for different values of *c*.

*c* |
inj |
surj |
cts |
mono |
bound |

(−∞,−2) |
✔ |
✗ |
✗ |
✗ |
✗ |

**−2** |
✔ |
✔ |
✗ |
✗ |
✗ |

(−2,−1) |
✗ |
✔ |
✗ |
✗ |
✗ |

−1 |
✗ |
✗ |
✗ |
✗ |
✔ to [0,1) |

(−1,0) |
✗ |
✔ |
✗ |
✗ |
✗ |

0 |
✔ |
✔ |
✔ |
✔ (<) |
✗ |

(0,+∞) |
✔ |
✗ |
✗ |
✔ (<) |
✗ |

where

inj = *injective* or *1-to-1* ()

surj = *surjective* or *onto* ()

cts = *continuous*

mono = *monotonic*:

(<) for strictly increasing ()

bound = *bounded range*

For *f* to be invertible (i.e. for *f* to have an inverse), *f* must be *bijective*, that is, both injective and surjective. There are only two values for which that is the case: . When *c*=0, *f* is just the identity function, and thus the case *c*=−2 is the only interesting case.

Please see also the question I raised on MathOverflow prior to writing this entry.

### Like this:

Like Loading...

*Related*

This entry was posted on Tuesday, 30 April 2013 at 23:00 and is filed under Contribution, Mathematics. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

## Leave a Reply