Can a semialgebraic function be continuously extended to the closure of its domain?

Question

Let $C$ be a bounded semialgebraic set and $F: C \to \mathbb{R}$ be a bounded and continuous semialgebraic function.

Is it true that $f$ can be continuously extended to the closure of $C$? I was not able to find a counter-example, but I believe I am missing something and unable to prove it.

Answer 1

Take $C = [-1,1]\setminus \{0\}$, and define: $$F(x)= \begin{cases} 1 &\text{if }x>0\\ 0&\text{if }x<0.\end{cases}$$ $C$ and $F$ are bounded and semialgebraic, and $F$ is continuous on $C$, but $F$ cannot be continuously extended to $\overline{C} = [-1,1]$.