1. ## Continuity Q

I know that function f(x) is continuous as f(a) as x-->a.

But why is f:= 1/x continuous?

But f(x) can't be defined at a x = 0.

I've heard that zero is not included in the domain, hence it is continuous.

Is that stating that f:= 1/x is in the domain of {x l (-inf,0) U (0,inf)} ?

But for 1/x^2, it's not continuous and it has the same characteristics as 1/x

Thank you!!

2. ## Re: Continuity Q

Originally Posted by si2136
I know that function f(x) is continuous as f(a) as x-->a.

But why is f:= 1/x continuous?

But f(x) can't be defined at a x = 0.

I've heard that zero is not included in the domain, hence it is continuous.

Is that stating that f:= 1/x is in the domain of {x l (-inf,0) U (0,inf)} ?

But for 1/x^2, it's not continuous and it has the same characteristics as 1/x

Thank you!!
A function is said to be continuous if it is continuous at every point in its domain.

There's usually an implicit assumption that the domain is R \ {0} for the functions you've listed.

Remember, "f(x) = 1/x" isn't a function, a function requires us to also specify the domain and codomain.

3. ## Re: Continuity Q

Ahh I see, that helps me so much haha thank you!

So 1/x as a function in high school was wrong

4. ## Re: Continuity Q

Originally Posted by si2136
Ahh I see, that helps me so much haha thank you!

So 1/x as a function in high school was wrong
Yep high school lied to us a lot about maths

5. ## Re: Continuity Q

Originally Posted by si2136
Ahh I see, that helps me so much haha thank you!

So 1/x as a function in high school was wrong
Depends on how you were taught. If taught more rigorously, you'd have been told that the definition of the function must include the specification of the domain, often implied, but if necessary, explicitly spelt out.

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