How to check if this function is continuous

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Homework Help Overview

The discussion revolves around the continuity of a function at a specific point, particularly at z=0. Participants are examining the limits of the function along different paths to determine continuity.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of finding different limits along various paths and question what this means for the continuity of the function at z=0. There is also exploration of what specific limit values would indicate continuity.

Discussion Status

The conversation includes attempts to clarify the conditions under which the function would be considered continuous. Some participants provide insights into the relationship between the limits and continuity, while others seek confirmation of their understanding.

Contextual Notes

There is mention of specific limit values and their implications for continuity, indicating a focus on the definitions and assumptions related to limits in the context of 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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