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)