functionsreal-analysis
I want an example for a function $f$ which is strictly increasing in some subset $S$ of $ℝ$ such that $f^{-1}$ is not continuous on $f(S)$.I came up with my own example but I need to verify it.
$$f(x)=x ; x∊[0,1]$$
$$f(x)=2x; x∊(1,2]$$
I would appreciate if some one could tell if my example is incorrect and if so provide a better example.Thank you
Here is a way to construct such a function.
EDIT v2 I've edited to use a strictly increasing function.
Here is a graphical depiction of the concept:
Just add the identity function, $\text{id}(x) = x$, to the Cantor function, $\text{c}$. The sum of continuous functions are continuous, and the sum of an increasing function with a strictly increasing one is strictly increasing.
As in the proof that $\text{c}$ is not absolutely continuous choose $\epsilon < 1$. For every $\delta > 0$ there is a finite pairwise disjoint sequence of intervals $(x_k,y_k)$ covering the zero-measure Cantor set with
$$ \sum_{k} |y_{k} - x_{k}| < \delta $$
And since the $\text{c}$ only changes on the Cantor set
$$\sum_{k} |\text{c}(y_{k}) - \text{c}(x_{k})| = 1$$
But
$$\begin{align} (\text{id}(y_{k}) + c(y_{k})) - (\text{id}(x_{k}) + c(x_{k})) &= (\text{id}(y_{k}) - \text{id}(x_{k})) + (c(y_{k}) - c(x_{k})) \\ &\ge c(y_{k}) - c(x_{k}) \end{align}$$
So a fortiori
$$\sum_{k} |(\text{id}(y_{k}) + c(y_{k})) - (\text{id}(x_{k}) + c(x_{k}))| \ge 1$$
Best Answer
The question does not have a positive answer, the only way to get discontinuity of the inverse or pseudo inverse is for f to be flat in some intervals. As pointed in the comments, if f itself has discontinuities in S then f(S) is going to be a collection of disjoint intervals such that the inverse in continuous in each one of them.