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A few questions on continuity.

MathematicalPhysicist

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1) let f:[a,b]->R be a continuous function, for every a<=t<=b we define:
M(t)=sup{f(x)|a<=x<=t} prove that M(t) is a continuous function on [a,b].
2) let f be a continuous function on [a,b], and we define that A={x in [a,b]| f(x)>=0} where f(a)>0>f(b).
i need to show that the supremum of A, s, that f(s)=0.
3) i need to prove/disprove that f(x)=arctan(x) is a uniform continuous on R.

for the first question, i think bacuase f is continuous then by wierstrauss theorem f achieves its max and min in the interval, so also M(t) is bounded, and the supremum of f(x) in a<=x<=t is also achieved in this interval and thus M(t) is continuous, i think this is basically the idea but i need to formalise this, anyone can help me on this?
and the second one, i showed that A is bounded and non-empty so it has got a supremum, now we have two options, or s in A or s isnt in A.
if s is in A, then f(s)>=0, so in oreder to show that f(s)=0, i tried to show ad absurdum that f(s)>0 leads to contradiction, but i didnt succeded in it, where did i go wrong?

also any help on the third question will helpful.
 

matt grime

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1) so your definition of continuous is what precisely? All you've done so far is show that it doesn't have any singularities.

2) seems easy enough, the sup is obvisously in A: f is continuous. We don't know where you went wrong since your attempted proof doesn't appear to be posted here. The method you cite will work - if f(s) is not zero there must be some t with f(t)>0 and t>s. Hint: t=s+delta/2, and f(t)=epsilon/2.
 

StatusX

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for the first question, i think bacuase f is continuous then by wierstrauss theorem f achieves its max and min in the interval, so also M(t) is bounded, and the supremum of f(x) in a<=x<=t is also achieved in this interval and thus M(t) is continuous, i think this is basically the idea but i need to formalise this, anyone can help me on this?
If f is continuous on a closed interval, then it's bounded on that interval. But the converse, that any such function is continuous, clearly does not hold (just take f(x)=sgn(x) on [-1,1]). Use the epsilon delta definition.

For the third, it might be easier to prove the more general result that any differentiable function whose derivative is bounded is uniformly continuous, and then prove arctan is such a function.
 

MathematicalPhysicist

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1) so your definition of continuous is what precisely? All you've done so far is show that it doesn't have any singularities.
so do you have any advice?
2) seems easy enough, the sup is obvisously in A: f is continuous. We don't know where you went wrong since your attempted proof doesn't appear to be posted here. The method you cite will work - if f(s) is not zero there must be some t with f(t)>0 and t>s. Hint: t=s+delta/2, and f(t)=epsilon/2.
but why the sup is obviously in A?
bacuse f is continuous means that the image of the interval [a,b] is also an interval, but how does this show that s is in A.
s might be greater than b and b isnt in A, or smaller than b, but still not in A.
i dont understand why it's obvious it's in A.
and for the method, if f(s) isnt zero, i.e f(s)>0, but why there must be some t such that f(t)>0, and t>s, is it also because f is continuous?
 

MathematicalPhysicist

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If f is continuous on a closed interval, then it's bounded on that interval. But the converse, that any such function is continuous, clearly does not hold (just take f(x)=sgn(x) on [-1,1]). Use the epsilon delta definition.
so you mean:
Ae>0,Ed>0,Ax in [a,b], |x-x0|<d => |f(x)-f(x0)|<e
(x0 is in [a,b])
i need to show that for every e>0 there exists d'>0 such that for every t in [a,b] (t0 in [a,b]) |t-t0|<d' then |M(t)-M(t0)|<e
but M(t)=sup{f(x)|a<=x<=t}
so M(t0)=sup{f(x)|a<=x<=t0}
because f is continuous: for every x in [a,t] which satisfies |x-x0|<d1
then |f(x)-f(x0)|<e/2 (where x0 is in [a,t])
also for every x in [a,t0] which satisfies |x-x0|<d2 (x0 is in [a,t0]) then |f(x)-f(x0)|<e/2
then: lets take d=min{d1,d2}
for |x-x0|<d
we have |M(t)-M(t0)|<=|M(t)-f(x0)|+|M(t0)-f(x0)|<e
is this correct?
 

matt grime

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so do you have any advice?
Yes, and I gave it - start with the definition of continuity, since you haven't acutally done so. (There might be a better way of doing this, but I don't see it off the top of my head.)


but why the sup is obviously in A?
I suppose it depends on how much analysis you know

bacuse f is continuous means that the image of the interval [a,b] is also an interval, but how does this show that s is in A.
s might be greater than b and b isnt in A,
That is impossible. A is a subset of [a,b], so its sup must be less than b.

or smaller than b, but still not in A.
i dont understand why it's obvious it's in A.
Because A is the inverse image of a closed set, hence closed. As I say, this might not be a result you know. However, let's do it from scratch - sup(A) is a limit point of A - there is a sequence x_n of elements in A tending to s by definition. f(x_n)=>0, so what can you say about f(s)?

and for the method, if f(s) isnt zero, i.e f(s)>0, but why there must be some t such that f(t)>0, and t>s, is it also because f is continuous?
Draw a picture. Then prove what you've seen. Have you met the Intermediate value theorem? I'm just using the epsilon delta definition of continuity
 

MathematicalPhysicist

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yes i met already the intermediate theorem.
so lets see if i understand correctly.
if f(s)>0 let t=s+d/2 for d>0, then: for |s-t|=d/2<d then |f(s)-f(t)|<e/2
let f(s)>e/2 then f(s)<f(t)+e/2 then 0<f(s)-e/2<f(t) so we have t in A, and t>s a contradiction to that of s being the supremum of A.
is this correct?
 

matt grime

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It appears to be written backwards. You should start with f(s) as being the epsilon. You're just plucking d from nowhere: don't. In particular d should not appear before the e: d depends on e.

The point is that if f(x) is strictly bigger than zero, there is a neighbourhood around x where f is always strictly bigger than zero. I'm sure this idea has come up many times in your studies so far.
 

MathematicalPhysicist

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yes the idea has come up.
so you mean i should write it as:
let e>0, there exists d>0, let t=s+d/2 then |s-t|=d/2<d because f is continuous, let's choose f(s)=e>0, then |f(s)-f(t)|<e
0=f(s)-e<f(t) so f(t) is bigger than 0 and so is in A.

i want just to clear something out, what i wrote in post 7, is just the idea, obviously when i write it on paper it will include the quantifiers and the order of epsilon and delta would be adequate.

btw, what about my appraoch to question 1, in post 5, where should i make any corrections if needed in my proof?
 

MathematicalPhysicist

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For the third, it might be easier to prove the more general result that any differentiable function whose derivative is bounded is uniformly continuous, and then prove arctan is such a function.
let me know if my proof is correct.
let f be a differnetiable function in (a,b) and suppose that the derivative f'(x) is bounded in that interval, then
|f'(x)|<=M, |lim(x->x0)[f(x)-f(x0)]/x-x0|<=M, then for some neibourhood (x0-d,x0+d) |f(x)-f(x0)|<=M|x-x0| bacuse f is differnetiable then it's also continuous then for every |x-x0|<d for some d (which is connected to e>0) we can choose d=e/M, and we get that f is uniformly continuous.

and, now for arctan(x), i need to show that it's derivative is bounded, and its derivative is 0<1/(1+x^2)<1.
is this correct?
 

StatusX

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|f'(x)|<=M, |lim(x->x0)[f(x)-f(x0)]/x-x0|<=M, then for some neibourhood (x0-d,x0+d) |f(x)-f(x0)|<=M|x-x0|

This doesn't quite follow. Just because the limit of a function as x->x0 is <=M doesn't mean any neighborhood of x0 has values <=M. For example, the limit as x->0 of f(x)=x is 0, which is <=0, but every neighborhood of 0 contains positive numbers. Instead, integrate the function, ie, start with:

[tex] |f(x)-f(x_0)|=\left| \int_{x_0}^x f'(x) dx \right| \leq \int_{x_0}^x |f'(x)| dx[/tex]
 

MathematicalPhysicist

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did i say any neighbourhood, i meant there exists a neighbourhood where the quetient is smaller than M.
anyway, i cant use integrals.
 

MathematicalPhysicist

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and i didnt undersatnd your example, if f(x)=x then f'(x)=1 which isnt smaller then zero
 

StatusX

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When I said it doesn't follow that any neighborhood has that property, I meant it doesn't follow that there exists a single one with the property. And I was just talking about limits in general, but if you need to use a derivative as an example, then take f(x)=x^2. Even though the derivative of x^2 is 0 at x=0, and so in particular is <=0 there, there is no neighborhood of 0 on which the derivative is <=0. Of course the result that |f(x)-f(x_0)|<=M|x-x0| is true, I'm just saying you haven't proven it yet.

If you can't use integrals, you'll need to at least use the fact that the derivative is <=M on an entire neighborhood of x0, not just at x0. Can you use the mean value theorem?
 

matt grime

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Can I suggest a different method for part 3?

Pick e>0. Now, there exists a constant K such that

0<pi/2 - arctan(x) < e for all x with |x| > K. Arctan is uniformly continuous on [-K,K], the rest you can fill in.
 

MathematicalPhysicist

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When I said it doesn't follow that any neighborhood has that property, I meant it doesn't follow that there exists a single one with the property. And I was just talking about limits in general, but if you need to use a derivative as an example, then take f(x)=x^2. Even though the derivative of x^2 is 0 at x=0, and so in particular is <=0 there, there is no neighborhood of 0 on which the derivative is <=0. Of course the result that |f(x)-f(x_0)|<=M|x-x0| is true, I'm just saying you haven't proven it yet.

If you can't use integrals, you'll need to at least use the fact that the derivative is <=M on an entire neighborhood of x0, not just at x0. Can you use the mean value theorem?
yes i can.
but how to use it here?
 

StatusX

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If |f(x)-f(x0)|>M|x-x0|, by the MVT, you can show there'd be some point c in (x,x0) with |f'(c)|>M, contradicting M as a bound for |f'|.
 

MathematicalPhysicist

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oh, i see you mean lagrange theorem, yes i learned it, and it will do just fine here.
 

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