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.
|−1||✗||✗||✗||✗||✔ to [0,1)|
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.