Background:
Suppose that we have a function which is a real-valued piecewise continuous algebraic function $f : \mathbb{R}^{n} \longrightarrow \mathbb{R}$. By that we mean that: $f(x) = \begin{cases} f_{1}(x) \mbox{ if }~ x \in S_{1} \\ f_{2}(x) \mbox{ if }~ x \in S_{2} \\ f_{3}(x) \mbox{ if }~ x \in S_{k} \end{cases} $
Where $x = (x_1,...,x_n)$. Here:
- Each $f_{i}$ is an real-value algebraic function, meaning there exits a polynomial $p$ such that $p(x,f(x)) = 0$
- Each $f_{i}$ is continuous on $S_{i}$, meaning that we only really need to worry about continuity in boundaries of the $S_{i}'s$
- $S_{1}\cup S_{2} \cup S_{3}$ is connected.
These arise when considering semi-algebraic functions.
Question: If $f$ has a connected graph, does that imply $f$ is continuous.
I understand that having a connected graph is not enough to imply continuity in general, as the topologist sine curve shows. I'm just hoping that the lack of oscillatory behavior in a function which is piecewise algebraic will help. Any pointers would be welcome.
Thanks in advance.