Choosing an Invertible and differential function with two singularities

Question

I want a function $f(x)$ with following properties:

1. maps from $(0, 1)$ to $(0, 1)$.

2. $f(x)$ is one-to-one (invertible).

3. The 1st derivatives $f'(0)$ and $f'(1)$ are $+\infty$

4. The mapping doesn't significantly warp $x$. The best case is $f(x) \approx x$, but this won't satisfy other properties

5. $f(x)$ is continuously differentiable.

   Ideally, $f(x), f'(x), f''(x), \ldots$ are all continuous and well defined for $x \in (0, 1)$.

6. both $f(x)$ and $f^{-1}(x)$ are not complicated.

Answer 1

Hint

$$f(x)={\sqrt{x}-\sqrt{1-x}+1\over 2}$$

whose graph is like this https://www.desmos.com/calculator/28isfuy3zz

also$$f^{-1}(x)=\begin{cases} {1\over 2}-\sqrt{{1\over 4}-4(x^2-x)^2}&,\quad x>{1\over 2} \\ {1\over 2}+\sqrt{{1\over 4}-4(x^2-x)^2}&,\quad x\le {1\over 2} \end{cases}$$