Intermediate Value Property for Discontinuous Functions

JasMath33
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Homework Statement


upload_2016-6-27_9-0-49.png
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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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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...
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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.
 
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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Screenshot_2016-06-28-13-20-07.png

IVP theorem
 
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JasMath33 said:

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 x_n and y_n such that \lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n = 0 but \lim_{n \to \infty} f(x_n) \neq \lim_{n \to \infty} f(y_n) then f is not continuous at zero.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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