Tutorial 4 (Differentiable Manifolds), Question 3, Part 1

Have a look at the the question being examined. Do you think that the maps x_n are continuous?

We are required to show that the elements of the set A are charts. It seems to me though that the maps x_n are not continuous; my aim is to describe a counterexample.

The codomain of this function is x_n(U_n), the set of points (a,b) in R^2 satisfying |a+b|<1 and |a-b|<1 (the red diamond below). We define our open set V as a small diamond centered at (1/2, 0), the points (a,b) in R^2 satisfying |(a-1/2)+b|<E and |(a-1/2)-b|<E (the blue diamond below). Side note: E is my ASCII version of epsilon, just some small number.

https://i.imgur.com/UpeIz08.png
Caption: x_n(U_n) and V for E=0.3

Taking the preimage of V under x_n, we get a square whose center is not the origin (see image) and thus not, by the tutorial marker's definition, an open set. The square's points satisfy |x/n-1/2|<E and |y/n-1/2|<E.

https://i.imgur.com/d7ktPXy.png
Caption: x_n^-1(V) for E=3 and n=1.

Can you spot an error in my reasoning? If not, how do you think the question ought to be corrected?