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## Homework Statement

Let [itex]f(z) = z^{\frac{1}{3}}[/itex] be the branch of the cube root function defined on [itex]\left[0, 2\pi\right)[/itex]. Show that f is not continuous for [itex]z_{0}[/itex] where [itex]z_{0}[/itex] is such that Re([itex]z_{0}[/itex]) > 0 and Im([itex]z_{0}[/itex]) = 0.

## Homework Equations

For a sequence of complex numbers, [itex] z_{n} = x_{n} + iy_{n} [/itex] converging to [itex] z_{0} = x_{0} + iy_{0} [/itex],

lim [itex] z_{n} = z_{0} [/itex] iff lim [itex] x_{n} = x_{0} [/itex] and lim [itex] y_{n} = y_{0} [/itex]

f is continuous if:

lim f([itex]z_{n}[/itex]) = f([itex]z_{0}[/itex]) when lim [itex] z_{n} = z_{0}[/itex]

## The Attempt at a Solution

Let [itex]z_{0} = a + ib [/itex], a>0, b = 0. Let [itex]z_{n} = x_{n} + iy_{n} [/itex]. So [itex] lim z_{n} = z_{0} [/itex] iff [itex] lim x_{n} = a [/itex] and [itex] lim y_{n} = 0 [/itex].

[itex] f(z_{n}) = e^{\frac{i}{3} [log(|z_{n}|) + arg(z_{n}) + 2k\pi)}, [/itex] for k = 0,1,2

[itex] f(z_{n}) = (\sqrt{x_{n}^{2} + y_{n}^{2}})^{\frac{1}{3}}[cos(\frac{arg(x_{n}+iy_{n})+2k\pi}{3})+i sin(\frac{arg(x_{n}+iy_{n})+2k\pi}{3})] [/itex]

But [itex] f(z_{0}) = a^{\frac{1}{3}}(cos(\frac{2k\pi}{3})+i sin(\frac{2k\pi}{3}), k = 0,1,2 [/itex] since [itex] z_{0} = a [/itex]

I'm not sure where to go from here. When I take the limit of [itex] f(z_{n}) [/itex], it seems to agree with the expression for [itex] f(z_{0}) [/itex], though I'm uncertain how to handle the argument function. Is the discontinuity because of the choice of branch? But how does that give rise to a problem? Let's say I let [itex] z_{n} [/itex] approach a point on the real axis from above and another version approach the same point from below. Do they give the same value for the argument in the limit, or do they approach different values for the angle, say 0 for the approach from above and 2pi for the approach from below? But I thought arg(z) was defined between [-pi/2, pi/2], so wouldn't they approach the same angle, 0, from either direction? I'm asking about argument because my only clue is that somehow approaching from different positions gives different arguments for the roots, some of which may be outside the chosen branch, but I can't figure out how to show this. Any help, advice, or leading questions would be most appreciated. Thank you so much.