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Is this map from $\mathbb{R}^2$ to $\mathbb{R}^2$ continuous?

Let $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be defined as follows
$(x, y)\mapsto (x-y, 2x-y)$
Is this map continuous?
I’ve been thinking about this for a while. I think it’s probably not. What is a counter example?
Also, can we say something about the weaker condition that $T$ is only homotopic to the identity map?

A:

The map $T$ is not defined everywhere in $\mathbb{R}^2$. There are points $x,y\in\mathbb{R}^2$ where $T(x)=T(y)$.
A continuous map $f:\mathbb{R}^2\to\mathbb{R}^2$ is defined to be homotopic to a constant map if for every $x\in\mathbb{R}^2$, there exists a homotopy $f_t$ of $f$ to the constant map $x\mapsto f(x)=x$.
The map $f(x,y)=(x-y,2x-y)$ is not homotopic to a constant map.
We can set $f_0(x,y)=x$ and $f_1(x,y)=x+y$.

A:

Consider the set $\{(x, y)\in \mathbb{R}^2: \, x+y=0\}$. This set is an ellipse. Its boundary consists of two line segments, one from the origin to $(-1, 1)$ and the other from the origin to $(1, -1)$.
Now consider the mapping $T(x, y)= (x+y, 2x+y)$. This mapping takes an ellipse to an ellipse https://thetalkingclouds.com/2022/06/07/custom-dark-mode-for-opera-crack-license-key-free-for-windows-2022-latest/

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