Let $A$ and $B$ subsets of $\mathbb R$ such that both have cardinality $\mathfrak c.$ I want to construct a noncontact function $f\colon A\to B$ by transfinite induction such that $f$ is continuous function. I know how construct non continuous function by transfinite induction. This is just an question come to my mind I do not know how it can be done.
Any help will be appreciated greatly
Not quite transfinite induction, but a complete answer nonetheless. If $A$ is disconnected, then you can write $A=A_1\amalg A_2$ for two non-empty subsets which are clopen in the subspace topology of $A$. Then, select two distinct elements $b_1,b_2\in B$; the function $$f:A\to B\\ f(a)=\begin{cases}b_1&\text{if }a\in A_1\\ b_2&\text{if }a\in A_2\end{cases}$$ is continuous.
If $A$ is connected, then $A$ is an interval, and it's all a matter of whether or not $B$ contains some interval with at least two points. If it doesn't, then by intermediate value theorem all continuous functions $A\to B$ must be constant.
On the other hand, if without loss of generality $(b_1,b_2)\subseteq B$ for some $b_1<b_2$, then you can send $A\to (b_1,b_2)$ with one of the usual continuous functions from an interval into another (not necessarily onto). Say, for instance, $$f(a)=b_1+\frac{b_2-b_1}\pi\left(\frac\pi2+\arctan a\right)$$