Two possibilities: (i) either F is not closed, (ii) F is not bounded.
If F is not closed then there exists so that . Now define .
This function is continous and unbounded since gets arbitrary close to .
If F is not bounded then for some define .
Certainly this is not bounded since gets arbitrary away from .
EDIT: Opalg beat me too it .
By the way I think it is possible to find a function , do you agree ?