Are Functions Continuous at a Cusp or Corner?

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In summary, functions are not differentiable at cusps or corners because a unique tangent cannot be drawn at these points. However, a function can still be continuous at a cusp or corner if the limit as x approaches that point exists, the function is defined at that point, and the limit equals the value of the function at that point. This means that a function can be defined at a cusp or corner and still be continuous. Additionally, there are functions that are everywhere continuous but nowhere differentiable, such as f(x)= |x|.
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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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  • #2
Yes, you can.
 
  • #3
On a side-note, you can construct functions that are everywhere continuous, but nowhere differentiable..
 
  • #4
Thanks

Thanks for responding! :smile:
 
  • #5
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.
 

FAQ: Are Functions Continuous at a Cusp or Corner?

What is continuity?

Continuity is a mathematical concept that describes the unbroken and connected nature of a function or graph. It means that, at every point on the graph, the function exists and has a defined value.

How do you determine if a function is continuous?

A function is considered continuous if it meets three criteria: 1) the function is defined at the point in question, 2) the limit of the function as x approaches the point is equal to the value of the function at that point, and 3) the limit exists for values approaching the point from both the left and right sides.

What is the difference between continuity and differentiability?

Continuity refers to the connectedness of a function, while differentiability refers to the smoothness of a function. A function can be continuous but not differentiable, meaning it has no breaks or jumps but has a sharp turn or corner. On the other hand, a function that is differentiable is also continuous.

Can a function be continuous at a point but not on an interval?

Yes, a function can be continuous at a specific point but not on an entire interval. This means that the function is smooth and unbroken at that point, but may have breaks or jumps at other points within the interval.

How is continuity used in real-world applications?

Continuity is used in many fields, including physics, engineering, and economics. For example, it is used in analyzing the flow of fluids, predicting the behavior of structures under different conditions, and modeling financial markets. It is also an important concept in calculus and is used to solve various mathematical problems.

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