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
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.