Show there exists a smooth (bump) function....

[MxwllsPersuasns]

Homework Statement

For a1 < a2 < b2 < b1 and c > 0 real numbers show that there exists
a smooth (bump) function f : R → R with the following property: f(x) = ...
- 0 if x ≤ a and x ≥ b
- c if a2 ≤ x ≤ b2
- monotonic increasing if a1 < x < a2
- monotonic decreasing if b2 < x < b1

Homework Equations

The Attempt at a Solution

So I'm not exactly sure what the question is asking. Is it asking me to construct such a function?

- If so then would I go about it by saying that c could be any constant like "5" or "2/3" and then go and simply give an example of a monotonic increasing function (like y = x) for that interval then give an example of a monotonic decreasing function (like y = -x) for that particular interval?

Or perhaps is the question asking me to show that a function like this is smooth?

- In which case we would essentially need to show that its a C-∞ function (at least all the examples I've seen say you need to show indefinite differentiation potential rather than just derivatives existing up to a certain, sufficient, degree.) However it doesn't appear that way since f(x) (given the example functions I've thrown in) would be twice differentiable, at most.

+ Unless of course then the implication is that we need to find monotonic functions which are also C-∞ functions in order to ensure the indefinite differentiability of f(x).

Any help anyone can provide about how to proceed or my proposed methods is extremely appreciated, I have this problem due for tomorrow morning and I'm just not quite sure what exactly needs to be shown here. Thanks in advance everyone, hoping for a lively and insightful conversation :)

 

[Ray Vickson]

Homework Statement

For a1 < a2 < b2 < b1 and c > 0 real numbers show that there exists
a smooth (bump) function f : R → R with the following property: f(x) = ...
- 0 if x ≤ a and x ≥ b
- c if a2 ≤ x ≤ b2
- monotonic increasing if a1 < x < a2
- monotonic decreasing if b2 < x < b1

Homework Equations

The Attempt at a Solution

So I'm not exactly sure what the question is asking. Is it asking me to construct such a function?

- If so then would I go about it by saying that c could be any constant like "5" or "2/3" and then go and simply give an example of a monotonic increasing function (like y = x) for that interval then give an example of a monotonic decreasing function (like y = -x) for that particular interval?

Or perhaps is the question asking me to show that a function like this is smooth?

- In which case we would essentially need to show that its a C-∞ function (at least all the examples I've seen say you need to show indefinite differentiation potential rather than just derivatives existing up to a certain, sufficient, degree.) However it doesn't appear that way since f(x) (given the example functions I've thrown in) would be twice differentiable, at most.

+ Unless of course then the implication is that we need to find monotonic functions which are also C-∞ functions in order to ensure the indefinite differentiability of f(x).

Any help anyone can provide about how to proceed or my proposed methods is extremely appreciated, I have this problem due for tomorrow morning and I'm just not quite sure what exactly needs to be shown here. Thanks in advance everyone, hoping for a lively and insightful conversation :)

Usually "smooth" means $C^1$, but maybe your book uses a different definition.

The problem does say "show that ...", so it gives you a choice, of either just proving existence somehow (perhaps using results from the book), or showing it by actually constructing an example. It really does seem to be up to you.

How you should proceed depends on what material you already have available from your textbook and/or course notes, and we have no idea what that is. Does your book already contain some examples of smooth functions that are, for example, zero for $x \leq 0$, 1 for $x \geq 1$ and smooth, monotonic on $\mathbb{R}?$ If so, you can just adapt that to your problem. If your book or notes does not contain such an example, you need to think about how you would construct one. (BTW: that is useful to do, because such functions are very often used in applications where we need to smooth out some curves, etc.)

 

[fresh_42]

Usually "smooth" means $C^1$, but maybe your book uses a different definition.

Usually smooth means $C^\infty$.
https://en.wikipedia.org/wiki/Smoothness

 

[andrewkirk]

Is it asking me to construct such a function?

It is not asking you to do that necessarily. It is conceivable that you could show the existence of such a function without constructing it. But in this case I think the easiest way to show it is to construct it.

Or perhaps is the question asking me to show that a function like this is smooth?

Yes, it's asking that. To construct the function you will need to make use of a simpler bump function, such as this one. What the problem statement does is tell you what characteristics the derivative of the answer function must have. Try to think of a way to combine two or more simple bump functions to create a function that has those characteristics.

By the way, the a and b below should be a1 and b1.

- 0 if x ≤ a and x ≥ b

 

[Ray Vickson]

Usually smooth means $C^\infty$.
https://en.wikipedia.org/wiki/Smoothness

Usually smooth means $C^\infty$.
https://en.wikipedia.org/wiki/Smoothness

Different sources have different definitions.
For example:
http://mathworld.wolfram.com/SmoothFunction.html
requires only continuous derivatives up to some order, not necessarily infinity.
Some respondents to
http://math.stackexchange.com/questions/472148/smooth-functions-or-continuous
also allow this; others do not.

However, perhaps I should have said "often" instead of "usually". In any case the OP should consult his textbook or course notes for the appropriate defintion.

 

[fresh_42]

I have also learned it as $C^\infty$ in my language and haven't heard of $C^1$ before.

 

[MxwllsPersuasns]

Hmm so sounds like the consensus here is it's valid to construct such a function and then show that it's a smooth function by proving the differentiability up to an acceptable point (which remains arbitrary at this point it seems -- being that C^∞ seems to be more difficult to achieve. Does that sound right?

P.S. I've heard of C¹ and C^∞ before but have seen different definitions of smoothness as well, there doesn't seem to be an accepted convention for it.

 

[andrewkirk]

it's valid to construct such a function and then show that it's a smooth function

Showing that will be very easy if you follow the method suggested in post 4, since the function you get will be the integral of a function that is itself smooth.

 

[MxwllsPersuasns]

Okay, when you say construct from a simpler bump function like the one in the link (where from [-1, 1] you have e^1/(1-x2)) what exactly do you mean? I'm thinking of writing out the curly bracket then literally just putting 0 for the appropriate interval and where it says "c" I was just going to put a number like 5 and then argue you can put any constant value, c there and then for the monotonic functions I was going to select some example monotonic functions (again, like y = x and y = -x) and just place them in the appropriate intervals but what you're saying is I need to use a bump function like that one or that one to construct... what exactly? Would I just give the function and then say that since we're dealing with positive constants that its e^(positive value) and then when we need it to be monotone decreasing just set it to e^(negative value) and then in the middle where it's constant just set it to e^(0)??

 

[andrewkirk]

I need to use a bump function like that one or that one to construct... what exactly?

As I said in post 4, you need to focus on the derivative of the function that you are going to give as an answer. Call that derivative function f. Here are some questions to help you think in the right direction:

1. What do we need the value of $\int_{a1}^{a2}f(x)dx$ to be?

2. Can we create a function that is a simple modification of a simple bump function (eg just multiplying by a constant and or adding a constant) whose integral over $[a1,a2]$ is equal to that value?

3. What do we need the value of $\int_{b2}^{b1}f(x)dx$ to be?

4. Can we create a function that is a simple modification of a simple bump function (eg just multiplying by a constant and or adding a constant) whose integral over $[b2,b1]$ is equal to that value?

5. Can we somehow put together the two functions, from Q2 and Q4, to make a smooth function that will have all the necessary requirements for $f$? How?

6. Having done 1-5, we just define our answer function $F$ as an appropriate definite integral of $f$.

 

[MxwllsPersuasns]

So I would imagine we need the integrals to both equal c. I'm not so sure about the modification that we would make to f. If we take f to be -2x/(1-x^2)*exp(-1/(1-x^2)) then I'm not exactly sure how to make that equal c over the selected intervals?

I was using the construction of the simple bump function and defining a1 = -b1 = -1 such that (using your notation -- so the answer function) F(±1) = 0 since we have e^-∞ I was trying to do the same by changing the numerator so that its e^(0) if I take c = 1 from a₂ to b₂ but I can't seem to make that happen.

 

[andrewkirk]

If we take f to be -2x/(1-x^2)*exp(-1/(1-x^2)) then I'm not exactly sure how to make that equal c over the selected intervals?

If you had been asked to find a function whose integral on the interval $[u,v]$ is 5, and you know a function $g$ such that $\int_u^v g(x)dx=10$, what might you do?