How can we describe the graph of $x^x$ for negative values?

The continuous curves that pass through the points satisfying $y = x^x$ on $x \in (-\infty, 0)$ are simply $$g(x) = \pm \frac{1}{|x|^{-x}}.$$

I would like to mention that your claim about extending a continuous function on a dense set, is not quite correct. For example consider $f(x)={1\over x}$ defined on $\mathbb{R}-\{0\}$. This is continuous on a dense set, but $f(0)$ cannot be defined to maintain continuity. You need the fact that the limit exists at every real number.