Why is $\log z$ continuous for $x\leq 0$ rather than $x\geq 0$? Explain why Log (principal branch) is only continuous on $$\mathbb{C} \setminus\{x + 0i: x\leq0\}$$ is the question.
However, I can't see why this is. Shouldn't it be $x \geq 0$ instead?
Thanks.

It all depends how you define your function $\log z$. Normally, the principal branch is defined by $\log(re^{i\theta}) = \log r + i\theta$ where $z = re^{i\theta}$ for some $\theta \in (-\pi, \pi]$, then you can check that it is not continuous for those $z$ with $\theta = \pi$, i.e. you approach from top and bottom gives you different values.
You should notice that $\log$ can be well-defined in many other cases. Suppose $f$ is a holomorphic function and $\Omega \subset \mathbb{C}$ is a simply connected region such that $f(z) \neq 0, \forall z \in \Omega$, then we can define $\log f(z) = \int_{z_0}^z\frac{f'(\zeta)}{f(\zeta)}d\zeta$, which is a well-defined continuous function on $\Omega$.
