In general there is no relation between the conditions of openness and continuity for functions between topological spaces.
For example, $f\colon (\mathbb{R}, \text{left ray topology})\to (\mathbb{R}, \text{discrete topology})$ given by $f(x)=x$ is open but not continuous.
My question if there is a function $(\mathbb{R}, \text{usual topology})\to (\mathbb{R}, \text{usual topology})$ which is open but not continuous?
And if so, what the conditions must be in this function to be open and not continuous?
The Conway base 13 function $f: \mathbb{R} \to \mathbb{R}$ is a rather extreme example: it is nowhere continuous but $f[(a,b)] = \mathbb{R}$ for all $a < b$ in $\mathbb{R}$, and this trivially implies that $f$ is open.