Let $(\Bbb R,+)$ be given the semi-open topology, i.e., the topological basis consists of open sets like $\{ [a,b)\}$.
Then show that,
$(i) \ g_1: \Bbb R \times \Bbb R \to \Bbb R$ given by $(x,y) \to xy$ is continuous,
$(ii) \ g_2: \Bbb R \to \Bbb R$ given by $g_2: x \to x^{-1}$ is not continuous.
My appraoch:
$(i)$ At first,
$g_1: \Bbb R \times \Bbb R \to \Bbb R$ and then $g_1: [a,b) \times [c,d) \to [u, v)$. But how does $g_1^{-1}([u,v))$ contains into the domain?
Also help me to show the $(ii)$.
As $0 \in [0,b)$ but what is $g_2^{-1}([0,b))$ ?
We will use the following definition of continuity: $f : X \to Y$ is continuous at $x \in X$ if for any basis set $V \subset Y$ containing $f(x)$, there exists an open set $U \subset X$ containing $x$ such that $f(U) \subset Y$. We say that $f$ is continuous if it is continuous at each $x$ in $X$.
Fix $(a,b) \in \mathbb{R} \times \mathbb{R}$ and an open set $[c,d) \subset \mathbb{R}$ containing $g_1((a,b)) = a + b$. Then $c \leq a + b < d$. In particular, this means that there exists $\epsilon > 0$ such that $a + b + \epsilon < d$.
If $$ (x,y) \in \left[a , a + \frac{\epsilon}{2} \right) \times \left[b , b + \frac{\epsilon}{2}\right) $$ Then $$ g_1((x,y)) = x + y \in [a + b, a + b + \epsilon) \subset [c, d) $$ Thus $$ g_1\left(\left[a , a + \frac{\epsilon}{2} \right) \times \left[b , b + \frac{\epsilon}{2}\right) \right) \subset [c,d) $$ This shows that $g_1$ is continuous at $(a,b)$. Since $(a,b) \in \mathbb{R} \times \mathbb{R}$ was arbitrary, this shows that $g_1$ is continuous.
Can you see what goes wrong for $g_2$?