Understanding Continuity: Exceptions to the Definition | Question on Continuity

[GreenPrint]

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

 

[Mark44]

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.

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.

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

 

[GreenPrint]

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!

 

[Mark44]

The context of the problem usually makes it clear what the domain for an equation is.

 

[Dick]

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?