Discontinous functions question.

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In summary, a function must meet three conditions to be continuous: it must be defined at a point, its limit must exist at that point, and the limit must be equal to the value of the function at that point. For the function f(x) = [x], which assigns an integer smaller than x to f(x), the instructor claimed that it is discontinuous for every integer x. However, two of the three requirements for continuity are actually met, as the limit exists in both directions and the function is defined for any x. The limit from below is equal to 0 and the limit from above is equal to 1, making the limit at x = 1 undefined. This shows that the function is not continuous at x =
  • #1
peripatein
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Hi,
For a function to be continuous, three conditions must be met - the function must be defined at a point x0, its limit must exist at that point, and the limit of the function as x approaches x0 must be equal to the value of the function at x0.
Now, assuming my function is f(x)=[x], which assigns an integer smaller than x to f(x). Thus, if 1≤x<2, f(x)=1.
The instructor claimed that this function is discontinuous for every integer x, which is perfectly clear just from looking at the graph of f(x), which is a step graph. He also mentioned that two of the three requirements for continuity are unmet in this case.
Which two requirements of the three above are unmet in this case?
 
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  • #2
peripatein said:
Hi,
For a function to be continuous, three conditions must be met - the function must be defined at a point x0, its limit must exist at that point, and the limit of the function as x approaches x0 must be equal to the value of the function at x0.
Now, assuming my function is f(x)=[x], which assigns an integer smaller than x to f(x). Thus, if 1≤x<2, f(x)=1.
The instructor claimed that this function is discontinuous for every integer x, which is perfectly clear just from looking at the graph of f(x), which is a step graph. He also mentioned that two of the three requirements for continuity are unmet in this case.
Which two requirements of the three above are unmet in this case?

For the limit to exist, the left- and right-side limits have to exist and be equal. You're dealing with a step function that has jumps at integer values. Take a look at the three conditions at these integer values.
 
  • #3
The instructor claimed that for a limit to exist at a point x0, the limit has to be equal both from above and below, unless the limit from one direction does not exist and then the limit at x0 DOES exist. Hence, that requirement is met for the step function as described above, I believe. Is it not?
Another requirement that is met is that the function is defined for any x0. Is it not?
So two requirements are already met and that alone disagrees with the instructor's claim that two conditions are unmet.
Could someone please clarify? What am I missing?
 
  • #4
For your function, what are
$$ \lim_{x \to 1^-} f(x)?$$
and
$$ \lim_{x \to 1^+} f(x)?$$

Do both exist? If so, are they equal?
 
  • #5
peripatein said:
The instructor claimed that for a limit to exist at a point x0, the limit has to be equal both from above and below, unless the limit from one direction does not exist and then the limit at x0 DOES exist. Hence, that requirement is met for the step function as described above, I believe. Is it not?
Another requirement that is met is that the function is defined for any x0. Is it not?
So two requirements are already met and that alone disagrees with the instructor's claim that two conditions are unmet.
Could someone please clarify? What am I missing?

No, the function f(x) = [x] does not have a limit as x → n, for every integer n. The limits exist in BOTH directions, however. (Why do these two statements not contradict each other?)

RGV
 
  • #6
Mark, is the limit from below equal to 0, whereas the limit from above is equal to 1?
 
  • #7
Mark44 said:
For your function, what are
$$ \lim_{x \to 1^-} f(x)?$$
and
$$ \lim_{x \to 1^+} f(x)?$$

Do both exist? If so, are they equal?

Do both limits indeed exist and whereas the first is equal to 0, the second is equal to 1?
May someone please clarify?
 
  • #8
peripatein said:
Do both limits indeed exist and whereas the first is equal to 0, the second is equal to 1?
May someone please clarify?

What is f(0.9)? What is f(0.99)? What is f(1.1)? What is f(1.01)? Can you see now what is happening?

RGV
 
  • #9
Well, as x approaches 1 from below f(x) is still 0 (or does that limit not exist?). When x approaches 1 from above, f(x) is 1. Is that correct?
 
  • #10
peripatein said:
Well, as x approaches 1 from below f(x) is still 0 (or does that limit not exist?). When x approaches 1 from above, f(x) is 1. Is that correct?
Yes. So
$$ \lim_{x \to 1^-} f(x) = 0$$
and
$$ \lim_{x \to 1^+} f(x) = 1$$

So does ##\lim_{x \to 1} f(x) ## exist?

Can you now answer the question about whether f is continuous?
 
  • #11
Yes, Mark. Thank you very much! :-)
 

1. What is a discontinuous function?

A discontinuous function is a type of mathematical function that has at least one point where the function is not defined or is not continuous. This means that the graph of the function will have a break or gap in it.

2. How do you identify a discontinuous function?

A discontinuous function can be identified by looking for points where the function is not defined or where there are abrupt changes in the graph. These points are called discontinuities.

3. What are the different types of discontinuities?

There are three main types of discontinuities: removable, jump, and infinite. Removable discontinuities occur when the function can be made continuous by redefining the value at that point. Jump discontinuities occur when the left and right limits at a point are different. Infinite discontinuities occur when the function approaches infinity at a certain point.

4. How do discontinuous functions affect real-world applications?

Discontinuous functions are often used in real-world applications to model situations where there are abrupt changes or breaks. For example, a discontinuous function may be used to model the stock market, where prices can suddenly jump or drop. They can also be used to model physical phenomena such as earthquakes or discontinuous phase transitions.

5. Can discontinuous functions be integrated?

Yes, discontinuous functions can be integrated, but it may require different techniques than continuous functions. For example, if a function has a jump discontinuity, the integral can be split into multiple integrals at each discontinuity point. However, if the function has an infinite discontinuity, the integral may not exist.

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