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# Continuity/Discontinuity help watch

1. Ok I'm kinda struggling with these quetsions any help would be appreciated.

Warwick Analysis II 2008 Q2b+c

Prove from the epsilon-delta definition, that continuity of f at c = 0 holds.

i) f : R -> R given by f(x) = 2x/(x+2) for f(x) < 0 and f(x) = x for x 0

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I don't know how to do this. I get the feeling any delta I pick will do but I really don't know how to write this up formally.
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[/latex]

ii) f : R -> R given by f(x) = 3xsin(1/x)
if x 0 and f(x) = 0 if x = 0

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I think I found this the easiest. Let . And so if we see that and so f is continuous at 0

Prove from the epsilon-delta definition, that continuity of f at c = 0 does not hold.

i) f : R -> R given by f(x) = (x-2)/10 if x 0 and f(x) = 0 if x = 0

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I'm thinking about the discontinuity definition here - i.e. there exists epsilon > 0 such that for all delta > 0 we have |x| < delta but |
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(x-2)/10| . BUT WHAT'S THE EPSILON?!

ii) f : R -> R given by f(x) = 3sin(1/x)
if x 0 and f(x) = 0 if x = 0

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No idea
2. (Original post by Maths Failure)
Prove from the epsilon-delta definition, that continuity of f at c = 0 holds.

i) f : R -> R given by f(x) = 2x/(x+2) for f(x) < 0 and f(x) = x for x 0
|2x/(x+2)|<|2x| when |x|<1

(Original post by Maths Failure)
Prove from the epsilon-delta definition, that continuity of f at c = 0 does not hold.

i) f : R -> R given by f(x) = (x-2)/10 if x 0 and f(x) = 0 if x = 0
Prove that the limits on both sides is -1/5.

(Original post by Maths Failure)
ii) f : R -> R given by f(x) = 3sin(1/x) if x 0 and f(x) = 0 if x = 0
Assume the limit is 0 and set up a contradiction using the definition.
3. (Original post by Lord of the Flies)
|2x/(x+2)|<|2x| when |x|<1
Ok cheers

(Original post by Lord of the Flies)
Prove that the limits on both sides is -1/5.
How does that help with delta-epsilon? By the way, I mistyped it. It should be:

f : R -> R given by f(x) = (x-2)/10 if x < 0 and f(x) = x if x 0

(Original post by Lord of the Flies)
Assume the limit is 0 and set up a contradiction using the definition.
I tried this but couldn't get anywhere. More help?
4. (Original post by Maths Failure)
How does that help with delta-epsilon? By the way, I mistyped it. It should be:

f : R -> R given by f(x) = (x-2)/10 if x < 0 and f(x) = x if x 0
Use to prove that the limits are different depending on which side we approach 0. By definition the function is discontinuous at 0.

(Original post by Maths Failure)
I tried this but couldn't get anywhere. More help?
Show that for any there will always exist such that and hence cannot be made arbitrarily small.

Hint:

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. In other words, show that there is always an integer such that for any fixed

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