Functional Analysis: Continuity and Convergence

Question

I'm working on the exercise from Joseph Muscat's Functional Analysis:

Find examples of continuous functions $f$ $(X = Y = \mathbb{R})$ such that

a) $f$ is invertible but $f^{-1}$ is not continuous

b) $f(x_n) \to f(x)$ in $Y$ but $(x_n)$ does not converge at all

c) $U$ is open in $X$ but $f(U)$ is not open in $Y$. However, functions which map open sets to open sets exist (find one) and are called open mapping.

I've been thinking about the question but couldn't come up with the answers/examples to the exercise. Wonder if someone could please help with the examples to the question.

Thank you!

Answer 1

b), c), take $f(x)=1$, then $(x_{n})=(1,0,1,0,...)$ is not convergent still $f(x_{n})=1\rightarrow 1$, and $U=(0,1)$ is open in ${\bf{R}}$ but $f(U)=f(0,1)=\{1\}$ is not open in ${\bf{R}}$.

Answer 2

a): If f is continuous and one-to-one then it is strictly increasing or strictly decreasing. In the first case $f^{-1}$ is also increasing. If it is not continuous then it will have a jump discontinuity at some point. But then its range contains no value between the left limit and the right limit at that point. But the range of $f^{-1}$ is $\mathbb R$. This contradiction shows that $f^{-1}$ is necessarily continuous. An identical argument holds for the second case.