I don't understand why do we need part a) (I can prove it though). Why do the following proof won't work?
Since $f$ is continuous on $R$, it is continuous at $a$, and so $\exists\delta>0\ \forall x \ |x-a|<\delta\implies f(x)>f(b)$.
So, we have $x\in(a,b)$, and $f(x)>f(b)$, which contradicts the statement that all points in $(a,b)$ are "shadow points".
Which leaves us with only possibility that $f(a)=f(b)$.