limits - Shouldn't this function be discontinuous everywhere?

I was thinking about single point continuity and came across this function. $$
f(x) = \left\{
\begin{array}{ll}
x & \quad x\in \mathbb{Q}\\
2-x & \quad x\notin \mathbb{Q}
\end{array}
\right.
$$ We know this function is continuous only at $x=1$ . But doesn't that contradict our whole idea of continuity? A function is continuous if we are able to draw the function without lifting our pen or pencil. But here both the pieces of the function exist at specific places, so we have to lift our pen. Shouldn't the function be discontinuous everywhere? Looks like a stupid doubt though.

Answer

More precisely, a function is continuous over an interval if we are able to draw the function without lifting our pen or pencil within that interval, in our intuition. This function is only continuous at one point.