How to check if this function is continuous

[MissP.25_5]

Hello.
The question is in the attached, together with my attempt. As you can see, I found the limit, but I don't know what each value means. If I have calculated the limits correctly, how do I know know if f(z) is continuous at 0 or not?

 

[benorin]

You have two different answers for the limit along different paths so the limit does not exist, therefore f(z) is not continuous at z=0.

 

[MissP.25_5]

You have two different answers for the limit along different paths so the limit does not exist, therefore f(z) is not continuous at z=0.

So, if no.1 and no.2 both had +/-1 as limits, then the function would be continuous at 0?

 

[benorin]

So, if no.1 and no.2 both had +/-1 as limits, then the function would be continuous at 0?

No, if one is positive 1 and the other is negative 1, they are still different and thus the limit d.n.e. So it's not continuous there.

 

[MissP.25_5]

No, if one is positive 1 and the other is negative 1, they are still different and thus the limit d.n.e. So it's not continuous there.

Got it, thanks! By the way, could you look at my thread entitled contour integral please?