to show continuity you must show that the inverse image of an open set in (r,discrete) is open in (r,normal) right?

So take for example the set {a}. This set is open in (r,discrete). while the inverse image is

\(\displaystyle f^{-1}(a)=\{x\in \mathbb R: f(x)=a\}=\{a\} \) which clearly is not open in (r,normal).

so f is cant be continuous wi those two topologies.

Take note that changing the topologies doesnt change the way the function acts, or the inverse image acts, the crucial point is whether the set and its inverse image are open or not.