Limit of an inverse function

A counter-example with $a=b=0$ and $\lim_{x\to a}f(x)=f(a).$

Let $(b_n)_{n\in \Bbb N}$ be a strictly decreasing real sequence with $b_1=1/2$ and with $\lim_{n\to \infty}b_n=0.$

For $x\le 0$ let $f(x)=x.$

For $n\in \Bbb N$ let f map $(b_{n+1},b_n]$ bijectively onto $(b_{n+1},b_n).$ And let $f(n)=b_n.$

Let $f$ map $(b_1,\infty)\setminus \Bbb N$ bijectively onto $(b_1,\infty).$

Then $f:\Bbb R\to \Bbb R$ is a bijection, and $\lim_{x\to 0}f(x)=0=f(0).$

But $(b_n)_{n\in \Bbb N}$ converges to $0$ while $f^{-1} (b_n)=n, $ so $f^{-1}(x)$ does not converge as $x\to 0.$

If $x=a$, consider the sequence $(a_{n})=(a+\frac{b-a}{2}\frac{1}{n})_{n\geq 1}$ and the sequence $(b_{n})(b+\frac{b-a}{2}\frac{1}{n})_{n\geq1}$. We define $f$ as follows $$f(x)=\begin{cases}b+\frac{b-a}{2}\frac{1}{2n-1}&\text{ if }x=a+\frac{b-a}{2}\frac{1}{n},\ n>0.\\ b+\frac{b-a}{2}\frac{1}{2n}&\text{ if }x=b+\frac{b-a}{2}\frac{1}{2n},\ n>0.\\2b-a+\frac{b-a}{2}\frac{1}{n}&\text{ if }x=b+\frac{b-a}{2}\frac{1}{2n-1},\ n>0.\\x+b-a&\text{ else}.\end{cases}$$ Note that $f$ is well-defined as for all $m,n\in\mathbb{N}$ we have \begin{align*}|a+\frac{b-a}{2}\frac{1}{m}-b-\frac{b-a}{2}\frac{1}{n}|&=|a-b+\frac{b-a}{2}\frac{n-m}{mn}|\geq|a-b|-|\frac{a-b}{2}||\frac{1}{m}-\frac{1}{n}|\\&\geq|a-b|-\frac{1}{2}|a-b|>0\end{align*} Using similar arguments we find that $$f^{-1}(x)=\begin{cases}a+\frac{b-a}{2}\frac{1}{n}&\text{ if }x=b+\frac{b-a}{2}\frac{1}{2n-1},\ n>0.\\b+\frac{b-a}{2}\frac{1}{2n} &\text{ if }x=b+\frac{b-a}{2}\frac{1}{2n},\ n>0.\\b+\frac{b-a}{2}\frac{1}{2n-1}&\text{ if }x=2b-a+\frac{b-a}{2}\frac{1}{n},\ n>0.\\x+a-b&\text{ else}.\end{cases}$$ is well-defined, and evidently $f^{-1}$ is the inverse of $f$. So $f$ is bijective.

Now let $\varepsilon>0$ and take $\delta=\min(\varepsilon,\frac{b-a}{2})$. For all $x\in\mathbb{R}$ with $|x-a|<\delta$ we have $|f(x)-f(a)|\leq|x+b-a-b|=|x-a|<\varepsilon$. So $\lim_{x\rightarrow a}f(x)=b$.

Furthermore note that $\lim_{n\rightarrow\infty}f^{-1}(b+\frac{b-a}{2}\frac{1}{2n})=b\neq a$, so $\lim_{x\rightarrow b}f^{-1}(x)\neq a$.

For $a=b$ one can actually (contrary to my previous claim) define a similar function. We define $$f(x)=\begin{cases}a+\frac{1}{2n}&\text{ if }x=a+\frac{1}{n},\ n>0.\\a+\frac{1}{2n-1}&\text{ if }x=a+2+\frac{1}{2n-1},\ n>0.\\a+2+\frac{1}{n}&\text{ if }x=a+2+\frac{1}{2n},\ n>0.\\x&\text{ else}.\end{cases}$$ Using similar arguments we find that $f$ is bijective $\lim_{x\rightarrow a}f(x)=a$, but $\lim_{n\rightarrow \infty}f^{-1}(a+\frac{1}{2n-1})=a+2\neq a$.