This question asked to construct one-to-one function from $(a,b)$ onto $[a,b]$. I know there is a function but it seems the question to define this function explicitly. How this can be done?
Edit I did first by transfinite induction But I was wrong. Thanks to user @ Arturo Magidin. He told me this wrong and he answered the question.
I would object to your attempt as not being a construction. It really amounts to nothing but saying "they both have the same cardinality, namely $\mathfrak{c}$, therefore there is a bijection between them."
To help you figure out how to give a constructive function, let me show you how to give an explicit bijection between $[0,1)$ and $(0,1)$:
Define the sequence $(a_n)_{n\geq 1}$ by $a_n = \frac{1}{n+1}$. Note that $a_n\in (0,1)$ for all $n\geq 1$.
Define $f\colon [0,1)\to (0,1)$ as follows: $$f(x) = \left\{\begin{array}{ll} \frac{1}{2} &\text{if }x=0;\\ a_{n+1} &\text{if }x=a_n\text{ for some }n;\\ x &\text{otherwise.} \end{array}\right.$$ Note that this is indeed a construction: I've told you exactly how to compute $f$ at any point, explicitly. If you give me an $x$, I can tell you exactly where it is mapped. Verify that $f$ is indeed a bijection.
Compare it with your definition: if I just said "find a bijection $g$ between $[0,1)$ and $\mathfrak{c}$; then find a bijection $h$ between $(0,1)$ and $\mathfrak{c}$; now define $f$ as $h^{-1}\circ g$" (which is essentially what you are doing), could you tell me what $f(\frac{1}{2})$ is? What $f(\frac{\sqrt{2}}{2})$ is? What $f(0.25)$ is? No; but with the function I gave above, you can tell me exactly what the values are at any point. The $f$ I gave first is constructive; the one in this paragraph is not constructive.