An example of a Semi-Continuous Function which is not continuous **Definition 1.** Let $X$ be a topological space and $A\subset X$. Then $A$ is semi-open if $A\subset cl[int(A)]$ (closure of the interior of $A$). Equivalently, $A$ is semi-open iff there exists an open set $G$ in $X$ such that $G\subset A\subset cl(G)$. **Definition 2.** Let $X$ and $Y$ be topological spaces. A function $f:X\to Y$ is semi-continuous on $X$ if $f^{-1}(B)$ is semi-open in $X$ for every open set $B$ in $Y$. **Example.** Let $X=Y=[0,1]$. Let a function $f:X\to Y$ defined as follows: $f(x)=1$ if $0\leq x\leq \frac{1}{2}$ and $f(x)=0$ if $\frac{1}{2}<x\leq 1$. The example shows that $f$ is not continuous on $[0,1]$. May I know some tips on how to show that $f$ is semi-continuous on $[0,1]$?

Since $f$ takes only two values, the inverse image if an open (or in fact any) subset of $Y$ is either $\emptyset$ or $[0,\frac12]$ or $(\frac12,1]$ or $X$. Show that each of these sets is semi-open in $X$ according to definition 1.