Intermediate Value Property

  • #1
21
1

Homework Statement


upload_2016-6-27_9-0-49.png
'
Here is the given problem

Homework Equations




The Attempt at a Solution



a. For part a, I felt it was not continuous because of the sin(1/x) as it gets closer to 0, the graph switches between 1 and -1. Then I felt it might be continuous, therefore I am not sure.

b. For part b, I felt it has the Intermediate Value Property (IVP), because I can do something with the IVT. Those were my thoughts and ideas.
 
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Answers and Replies

  • #2
BiGyElLoWhAt
Gold Member
1,570
116
Can you show that the limit as approached positively is different from that as approached negatively? Or that it's different than f(0)? So in other words,
##\lim_{x \to 0} f(x) \neq f(0)##?

For b, I'm not really sure. I thought one of the requirements was that f(x) was continuous...
**
Perhaps this will be of some use.
http://math.stackexchange.com/quest...te-value-property-and-discontinuous-functions
Looks like you need to look at the derivatives near zero.
 
  • #3
134
12
For part a you would have to do what BiGyElLoWhAt suggested. For b I believe you would have to prove that the function in either monotone increasing or decreasing. IVP says that for any x value between two other x values, the y value will be in between the y values for the other two x values.
 
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  • #4
134
12
Screenshot_2016-06-28-13-20-07.png

IVP theorem
 
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  • #6
pasmith
Homework Helper
1,967
599

Homework Statement


View attachment 102545'
Here is the given problem

Homework Equations




The Attempt at a Solution



a. For part a, I felt it was not continuous because of the sin(1/x) as it gets closer to 0, the graph switches between 1 and -1. Then I felt it might be continuous, therefore I am not sure.
If there exist sequences [itex]x_n[/itex] and [itex]y_n[/itex] such that [itex]\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n = 0[/itex] but [itex]\lim_{n \to \infty} f(x_n) \neq \lim_{n \to \infty} f(y_n)[/itex] then [itex]f[/itex] is not continuous at zero.
 

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