Really Basic Question regarding Continuity

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In summary: Of course, that means the new function is NOT the same as the old one. It is the same except at x= 0- the "removable discontinuity" has been "removed".
  • #1
Mathguy15
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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.
 
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  • #2
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.
 
  • #3
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.
 
  • #4
Alrighty, thanks!
 
  • #5
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.
 

What is continuity?

Continuity is a concept in mathematics that refers to the smoothness and unbrokenness of a function or a curve. It means that there are no sudden jumps or holes in the graph of the function.

What is the difference between continuity and differentiability?

Continuity and differentiability are closely related concepts, but they are not the same. Continuity refers to the smoothness of a function, while differentiability refers to the existence of a derivative at a specific point. A function can be continuous but not differentiable, but if a function is differentiable, it must also be continuous.

How can we determine if a function is continuous?

A function is continuous if it satisfies the three conditions of continuity: the function must be defined at that point, the limit of the function at that point must exist, and the limit must equal the value of the function at that point. If all three conditions are met, the function is continuous at that point.

What is the Intermediate Value Theorem?

The Intermediate Value Theorem is a theorem in calculus that states if a continuous function has two different y-values at two points, then it must also take on every y-value in between those two points. This theorem is useful for finding roots or zeros of a function.

Can a function be discontinuous at a single point?

Yes, a function can be discontinuous at a single point. This means that the function is continuous everywhere except at that specific point. It can have a hole, a jump, or an infinite discontinuity at that point.

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