Give an example of continuously differentiable function that satisfy some properties

Question

Can you give an example of continuously differentiable function that satisfy this properties:

Let $\{(a_i, b_i)\}_{i=1}^{\infty}$ be the family of open subintervals of $[0,1]$ with rationals endpoints. For each $i\geq 1$, let

$$g_i:[a_i,b_i]\to [0,1]$$

be a continuously differentiable function such that

1. $g_i(a_i)=1$, $g_i(b_i)=0$,
2. $\lim_{x\to a_i^+} g_i^{\prime}(x)=0$,
3. $\lim_{x\to b_i^-} g_i^{\prime}(x)=0$.

I have tried to make some examples, this is my attempt:

1. $g_i(x)=\frac{b_i-x}{b_i-a_i}, a_i\leq x \leq b_i$. But, this function not satisfy properties poin (3) and (4).
2. \begin{align*} g_i(x)=\begin{cases} 1,& a_i\leq x < \frac{a_i+b_i}{2}\\ \frac{16x-4a_i-12b_i}{a_i-b_i},& \frac{a_i+b_i}{2}\leq x < \frac{a_i+3b_i}{4}\\ 0, & \frac{a_i+3b_i}{4}\leq x \leq b_i \end{cases} \end{align*} This function satisfies properties poin (1)-(4), but this is not continuously differentiable.

Finally, I got stuck on finding an example of that function. Any helps will be appreciated.

Answer 1

Hint. There are 4 conditions so try a polynomial with 4 c0-efficients, i.e. a cubic $$g(x)=\frac {x-b}{a-b}\cdot (\,A(x-a)^2+B(x-a)+1\,).$$