Need Help Determining Continuity of Functions

In summary, the first function is continuous at x = -1 because it exists for all x such that the limit exists. The second function is continuous at x = 1 because it exists for all x such that the limit exists. The third function is continuous at x = 2 because it exists for all x such that the limit exists. The fourth function is not continuous at x = 3 because it does not exist for that value of x.
  • #1
CJ256
13
0

Homework Statement



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Homework Equations



Ability to graph functions.
Essential Discontinuity:
Jump discontinuity or Infinite discontinuity

The Attempt at a Solution



First Question:

After plugging in 2 for every equation and getting a result that was greater than 0, I determined that the function was continuous and that the type of discontinuity is Essential (Infinite) discontinuity. The reason why I chose infinite is because when I drew the graph on my TI-84 Plus it didn't seem to have an empty point and all my points that I tested were filled. I don't understand how graph them by hand. I usually have no problem solving this type of questions when I have a graph, but when I have to make my own I really struggle.

Second Question:

After testing a couple of values (I tested, -1, 0, 1 as possible values of A) I determined that the answer to the question is all negative values could be values of A. I really struggled with this because I did not understand what the question really asked me. I kind of tried to satisfy the equations and once I saw that it did I decided that that was the answer. I know this question is wrong so any help would be really appreciated.
 
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  • #2
Do you have a definition for a function being continuous a point ?

If so, what is that definition?
 
  • #3
SammyS said:
Do you have a definition for a function being continuous a point ?

If so, what is that definition?

Yep. A function f(x) is continuous at x = c if, as x approaches c as a limit, f(x) approaches f(c) as a limit or in other words this:

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  • #4
choboplayer said:
Yep. A function f(x) is continuous at x = c if, as x approaches c as a limit, f(x) approaches f(c) as a limit or in other words this:

076.gif
So, first of all [itex]\displaystyle \lim_{x\,\to\,c}\,f(x)[/itex] must exist. If it exists, then it must be equal to f(c).

First Problem: What is [itex]\displaystyle \lim_{x\,\to\,2}\,f(x)\,?[/itex]

How do you determine whether or not this limit exists?

Second Problem: [itex]\displaystyle \lim_{x\,\to\,3}\,f(x)\,?[/itex]

How do make sure this limit exists?

How do make sure this limit is equal to f(3)?

What is f(3)?
 
  • #5
SammyS said:
So, first of all [itex]\displaystyle \lim_{x\,\to\,c}\,f(x)[/itex] must exist. If it exists, then it must be equal to f(c).

First Problem: What is [itex]\displaystyle \lim_{x\,\to\,2}\,f(x)\,?[/itex]

How do you determine whether or not this limit exists?

Second Problem: [itex]\displaystyle \lim_{x\,\to\,3}\,f(x)\,?[/itex]

How do make sure this limit exists?

How do make sure this limit is equal to f(3)?

What is f(3)?

Well for the first one if I plug in 2 where I have x all the equations are true except the x^3-3 so does that mean that it is not continuous even though it in the second equation 2=2?

For the second one I still have no clue where to start.
 
  • #6
Ok so for the first problem, I got that the function exists because of the piece wise function two of the functions equal 5 but I still need help with the second question
 
  • #7
choboplayer said:
Ok so for the first problem, I got that the function exists because of the piece wise function two of the functions equal 5 but I still need help with the second question
It's not asking if the function exists. It's asking if the limit exists.


It's quite clear from Post #5, that you don't understand the piecewise definition of a function. What are each of the following for the first function?
f(-1) =   ?  

f(0) =   ?  

f(1) =   ?  

f(1) =   ?  

f(1.9) =   ?  

f(2) =   ?  

f(2.1) =   ?  

f(3) =   ?  

f(4) =   ?  
 

FAQ: Need Help Determining Continuity of Functions

What is continuity of a function?

Continuity of a function refers to the idea that a function is unbroken or connected at every point in its domain. In other words, there are no sudden jumps or gaps in the graph of the function.

How can I determine if a function is continuous?

In order for a function to be continuous, it must satisfy three conditions: 1) the function must be defined at the point in question, 2) the limit of the function at that point must exist, and 3) the limit must be equal to the value of the function at that point. If all three conditions are met, the function is continuous.

What is the difference between continuity and differentiability?

Continuity and differentiability are related concepts, but they are not the same. A function is continuous if it is unbroken at every point in its domain, while a function is differentiable if it has a well-defined derivative at each point in its domain.

Can a function be continuous but not differentiable?

Yes, a function can be continuous but not differentiable. This can occur at points where the function has a sharp turn or a corner in its graph, as the derivative cannot be defined at these points.

How can continuity of a function affect its behavior?

A continuous function has a smooth and predictable behavior, which makes it easier to analyze and understand. In contrast, a discontinuous function can have unexpected jumps or breaks in its graph, making it more difficult to study and interpret.

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