Let $X=[0,1]\times [0,1]$ and $Y=[2,3]\times [0,1]$ be a subspaces from Moore plane. Define function $f,g,h\colon X\to Y$ by $f(x,y)=(x+2,y^2)$, $g(x,y)=(x+2,1-y)$, and $h(x,y)=(x+2,1-y^2).$ I want to check the if they are homeomorphism or not?
My attempt: For $g$, let $A=\{(x,1)\colon x\in[0,1]\}$ is compact but $g[A]=\{(x,0)\colon x\in[2,3]\}$ is not compact since it's topology is discrete topology. So, it is not homomorphism. The same reason works for $h$. For $f$, I think, it is homeomorphism but I did not see the easiest way to show that.
Attempt to answer my question but it might be wrong:
I am thinking to define $f_1\colon [0,1]\times (0,1]\to [2,3]\times [0,1]$ and $f_2\colon [0,1]\times \{0\}\to [2,3]\times [0,1].$ Put $f:=f_1\cup f_2$. Notice that $f_1$ is with Euclidean topology and $f_2$ with discrete topology. Clearly, $f_2$ is a continuous function. $f_1$ is continuous function, is it true $f_1$ continuous function?
$f$ is objection. In the same way, we could check $f^{-1}$ is a continuous function.