Are Functions Continuous at a Cusp or Corner?

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Homework Help Overview

The discussion revolves around the continuity of functions at points known as cusps or corners, particularly in the context of differentiability. The original poster is exploring the definitions and conditions for continuity in relation to these specific points on a function's graph.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to clarify the conditions under which a function can be continuous at a cusp or corner, referencing the definitions of continuity. Some participants affirm the possibility of continuity at these points, while others introduce examples of functions that are continuous but not differentiable.

Discussion Status

The discussion is active, with participants providing affirmations and examples related to the continuity of functions at cusps or corners. There is an exploration of different perspectives on the topic, but no explicit consensus has been reached.

Contextual Notes

Participants are considering the definitions of continuity and differentiability, and how they apply to specific functions, such as f(x) = |x|, which has a cusp at x = 0. The implications of these definitions are being examined without resolving the broader questions about continuity at cusps or corners.

Rozy
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I'm just working through some differentiability questions and have a quick question - are functions continuous at a cusp or corner? I know that functions are not differentiable at cusps or corners because you cannot draw a unique tangent at these points, but I'm not sure about continuity. From my understanding, a function is continuous if at point a if

1. The limit as x approaches a exists
2. f(a) exists or is defined
3. the limit of x approaching a = f(a)

Can you define the function at a cusp or corner and thus have a continuous function?

Thanks!
Rozy
 
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Yes, you can.
 
On a side-note, you can construct functions that are everywhere continuous, but nowhere differentiable..
 
Thanks

Thanks for responding! :smile:
 
For example f(x)= |x| has a "cusp" or corner at x= 0. Of course, f(0)= |0|= 0 so the function is certainly define there- and continuous for all x.
 

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