How to
(1) obtain some examples for an $f$ function and/or
(2) characterize the whole set of $f$ functions
which has the following properties given $m$ ($\leftarrow$ the median) that $0 < m < 1$ ($\leftarrow$ the average):
- $f\colon [-1, 1] \mapsto \mathbb{R}_{\ge0}$
- differentiable
- monotonically increasing
- $f(0) = m$
- $\int_{-1}^1 f(x) = 2$
For $m = 1$ we would have $f(x) = 1$ or $f(x) = x + 1$ as a trivial solution.
For the non-differentiable case we would have
$$f(x) = \begin{cases}m, & \text{if $x <= 0$}\\ 2-m, & \text{if $x > 0$}\end{cases}$$
as a trivial solution.
I would be happy with some examples to some values of $m < 1$ (e.g. $m=\frac12$).
Any additional constraint can be applied if it makes the task easier (e.g. the shape of $f$ should resemble to a specific right skewed distribution).