Understanding Continuity: Exceptions to the Definition | Question on Continuity

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In summary, the conversation is about the definition of continuity for a function. The book states that a function f is continuous at a number a if the limit of f(x) as x approaches a is equal to f(a). The person speaking is questioning if there are any exceptions to this definition or if it is completely wrong. They bring up a previous experience where they were told an equation had no solutions, but later found out that it did. They also mention the possibility of the complex logarithm being an exception. Ultimately, they just want to confirm if the book's definition is correct.
  • #1
GreenPrint
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Homework Statement



Ok my book tells me

A function f is continuous at a number a if

lim x->a f(x) = f(a)

and I'm not buying it

Like sure it makes sense but I'm wondering if someone can tell me the exceptions to this definition or if it's just completely wrong

you know like sort of like how when I was told many years ago

cox(x)=-2 had no solutions

only to find out it did...

like so can someone tell me if the definition my book gives me is strictly speaking correct... please tell me all exceptions or if it's wrong like for example maybe the complex logarithm maybe?

Thanks

Homework Equations


The Attempt at a Solution

 
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  • #2
GreenPrint said:

Homework Statement



Ok my book tells me

A function f is continuous at a number a if

lim x->a f(x) = f(a)

and I'm not buying it

Like sure it makes sense but I'm wondering if someone can tell me the exceptions to this definition or if it's just completely wrong
There are no exceptions.
GreenPrint said:
you know like sort of like how when I was told many years ago

cox(x)=-2 had no solutions
Why are you dragging this up again? It was explained to you that equations have domains on which they are defined. If x is a real number, then cos(x) = -2 has no solutions. If x is a complex number, then this equation has solutions.
GreenPrint said:
only to find out it did...

like so can someone tell me if the definition my book gives me is strictly speaking correct... please tell me all exceptions or if it's wrong like for example maybe the complex logarithm maybe?

Thanks
 
  • #3
Thanks... I just wanted to make sure =). Yes I know they have domains but when their not specified one cannot assume that x is in the set of reals, especially when you were taught it about other number systems...

I just wanted to make sure and thanks!
 
  • #4
The context of the problem usually makes it clear what the domain for an equation is.
 
  • #5
GreenPrint said:

Homework Statement



Ok my book tells me

A function f is continuous at a number a if

lim x->a f(x) = f(a)

and I'm not buying it

Like sure it makes sense but I'm wondering if someone can tell me the exceptions to this definition or if it's just completely wrong

you know like sort of like how when I was told many years ago

cox(x)=-2 had no solutions

only to find out it did...

like so can someone tell me if the definition my book gives me is strictly speaking correct... please tell me all exceptions or if it's wrong like for example maybe the complex logarithm maybe?

Thanks

Homework Equations





The Attempt at a Solution


What you quoted is the DEFINITION of continuity. Not buying a definition is a little strange. Do you have a different definition of continuity?
 

FAQ: Understanding Continuity: Exceptions to the Definition | Question on Continuity

What is continuity?

Continuity is a mathematical concept that describes the uninterrupted behavior of a function over a given interval. In simpler terms, it means that the graph of a function can be drawn without any breaks or holes.

What are the exceptions to the definition of continuity?

There are three main exceptions to the definition of continuity: removable discontinuity, jump discontinuity, and essential discontinuity. These occur when there is a break, hole, or infinite jump in the graph of a function, respectively.

How can I identify continuity or discontinuity in a function?

To identify continuity or discontinuity in a function, you can use the three-part definition of continuity. First, the function must be defined at the point in question. Second, the limit of the function at that point must exist. And third, the limit must be equal to the function value at that point.

Can a function be continuous at one point but discontinuous at another?

Yes, a function can be continuous at one point but discontinuous at another. This is because continuity is a local property, meaning it only applies to a specific point on the graph of a function. A function can be continuous at some points and discontinuous at others.

Why is continuity important in mathematics and science?

Continuity is important in mathematics and science because it allows us to make predictions and understand the behavior of functions. It also helps us to analyze real-world phenomena and make accurate models and approximations. Additionally, continuity is a fundamental concept that is used in calculus and other branches of mathematics.

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