Curved mirror vs flat mirror If I stand in front of a (horizontally) concave mirror at a distance equal to twice the focal distance, I should be able to see my reflection inverted. That is, if I lift my left hand, the reflection will lift the hand to my right. So basically it would be like a normal flat mirror, except that it doesn't invert left and right.
Now imagine that I can change the curvature of the mirror, making the focal distance progressively bigger, but I stay at twice the focal distance, when the focal distance becomes infinite (also placing me at infinity), it will have become a flat mirror but that doesn't flip my image.
So basically this thought experiment tells me that if you stand infinitely far in front of a flat mirror, your image won't be reversed. Is this physically plausible, a mathematical artifact/loophole or just plain stupid?
 A: I'd say the answer is, a very weakly concave mirror will act like a flat mirror if you are much closer to it than its focal length, and like a concave mirror if you are farther than its focal length.  The flatter it gets, the larger is the domain where it acts flat.  If the mirror does not have a constant curvature, then different parts of your reflection will correspond to different focal lengths, so the farther away you get, you will first pass outside the focal lengths of some parts and not others. So I think you are basically right-- the farther from a mirror you stand, the greater the chance you will be beyond the focal length of whatever piece of the mirror matters for forming some part of your reflection, so at some point you should expect the mirror to act as though it was curved and not flat.  But for a real mirror, this may produce a very fractured image, as some parts of your reflection make that transition before others do, and indeed some parts might even look like they are reflected from a convex mirror while others look concave.  So I think your basic point is well taken-- mirrors only work like mirrors out to a given distance that depends on how flat they are.
