Let $f$ be a function on $\mathbb R$ that satisfies the following property $$ \text{For every} \ a,b\in\mathbb R \ \text{and for every perfect set }\ K\ \text{between} \ f(a)\ \text{and} f(b), f(a)< K< f(b), \text{there exists a perfect set } \ C \ \text{and} \ a<C<b \ \text{such that}\ f[C]\subset K$$ It is general case from the intermediate value property and it is not hard to see every continuous function has the above property. My question is does $id\cdot f$ have the property above when $id$ the identity function and $f$ have the same property.
Definition. $A\subset\mathbb R$ is called a perfect set if $A$ is closed and contain no isolated points. For example, Cantor set is a perfect set.