I know how I can construct a function such that $f^{-1} (y)$ has size less than continuum actually countable.
Here is the proof,
Define a relationship as following $$x\sim y \ \text{iff} \ x-y\in\mathbb Q(x,y\in\mathbb R)$$. the equivalence classes have the form $[r]=r+\mathbb Q$ and clear they are countable dense and pairwise disjoint and $\mathbb R=\bigcup_{r\in\mathbb R} [r]$.
Let $\{B_r:r\in\mathbb R\}$ be partition of $\mathbb R$ to continuum many dense sets ( as the equivalence classes defined above) we define the function $f:\mathbb R\to \mathbb R$ by $f(x)=r$ for all $x\in B_r.$ It is that clear $f^{-1} (y)$ has size less than continuum actually countable. My question is if I want to construct a vector space of real-valued functions ($f:\mathbb R\to \mathbb R$) with cardinality continuum such that each function satisfies the property $f^{-1} (y)$ has size less than continuum actually countable for all $y\in\mathbb R$. I spent a lot of time but I did not get it. Any help will be appreciated.