How to check if this function is continuous

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To determine if a function f(z) is continuous at z=0, one must evaluate the limits along different paths. If the limits yield different values, as shown in the discussion, the limit does not exist, indicating that f(z) is not continuous at that point. Even if both limits approach the same absolute value but differ in sign, the limit still does not exist, confirming discontinuity. Therefore, continuity at z=0 requires identical limits from all paths. The discussion emphasizes the importance of consistent limit values for establishing continuity.
MissP.25_5
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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?
 

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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.
 
benorin said:
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?
 
MissP.25_5 said:
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.
 
benorin said:
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?
 

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