Really Basic Question regarding Continuity

[Mathguy15]

Hello,

I was reading through some lecture notes on Single-Variable Calculus, and the teacher gave this definition of continuity:

"A function f is called continuous at a point p if a value f(p) can be found such
that f(x) → f(p) for x → p. A function f is called continuous on [a, b] if it is
continuous for every point x in the interval [a, b]."

Now, I thought this meant the function actually had to be defined at p, but the teacher says that the function 1/log(|x|) is continuous at 0. Of course, log(|x|) isn't defined at 0. So, I have to ask the question, Does a function have to be defined at a point to be continuous at that point?

Thanks,
Mathguy

PS: I have found other definitions that say f(p) must exist.

 

[micromass]

Your teacher has some weird terminology. Usually, the function must always be defined at p in order for the function to be continuous.

Of course, often the function is not defined at p, but a value of f(p) can be given. Strictly speaking, we don't talk about a continuous function then. Such a thing is usually called a removable singularity. The function that is continuous is actually the unique extension of f at p.

 

[Vargo]

What micromass said. It is unusual for a definition of continuity.

Basically, it is the usual definition plus a stipulation that you can fill in gaps in the domain anywhere the limit exists but at which the function is not naturally defined by the formula. A more trivial example would be x^2/x. It is somewhat ambiguous whether 0 is in the domain of this function or not. Personally, I would say that the reasonable answer is that this is the same as the function x which is defined everywhere. So basically, your prof is saying the same sort of thing. If there is a defect in the formula that leaves out points, then we should automatically fill them in if possible.

 

[Mathguy15]

Alrighty, thanks!

 

[HallsofIvy]

According to you your teacher

gave this definition of continuity:

"A function f is called continuous at a point p if a value f(p) can be found such
that f(x) → f(p) for x → p. A function f is called continuous on [a, b] if it is
continuous for every point x in the interval [a, b]."

Since that requires that f(x)→ f(p), that definition requires that f(p) exist. By that definition, the function f(x)= 1/log(|x|) is NOT continuous at x= 0. We would say it has a "removable discontinuity" there. The discontinuity can be "removed" by defining f(0) to be 0.