There are plenty of continuous functions such that
. Besides the trivial examples
and
, one can take any equation
that is symmetric in
and has a unique solution for one variable in terms of the other. For example:
leads to
.

I can’t think of an explicit example that is also differentiable, but implicitly one can be defined by , for example. In principle, this can be made explicit by solving the cubic equation for
, but I’d rather not.

At the time of writing, I could not think of any diffeomorphism such that both
and
have a nice explicit form. But Carl Feynman pointed out in a comment that the hyperbolic sine
has the inverse
which certainly qualifies as nice and explicit.
Let’s change the problem to . There are still two trivial, linear solutions:
and
. Any other? The new equation imposes stronger constraints on
: for example, it implies
But here is a reasonably simple nonlinear continuous example: define
and extend to all by
. The result looks like this, with the line
drawn in red for comparison.

To check that this works, notice that maps
to
, which the function
maps to
, and of course
.
From the plot, this function may appear to be differentiable for , but it is not. For example, at
the left derivative is
while the right derivative is
.
This could be fixed by picking another building block instead of , but not worth the effort. After all, the property
is inconsistent with differentiability at
as long as
is nonlinear.
The plots were made in Sage, with the function f define thus:
def f(x): if x == 0: return 0 xa = abs(x) m = math.floor(math.log(xa, 2)) if m % 2 == 0: return math.copysign(2**(m + xa/2**m), x) else: return math.copysign(2**(m+1) * (math.log(xa, 2)-m+1), x)
Forward and inverse hyperbolic sine both can be expressed in an explicit form, taught in high school. Is that close enough to a “nice” invertible diffeomorphism?
sinh certainly qualifies, I didn’t think of hyperbolic functions when writing this. Updated the post. Thanks!