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I want to know if there exists an space for time-limited bounded continuous functions (at least one-variable ones), like the space of Bump functions $C_c^\infty$, but for functions that doesn't have to start or finish at $f(t_0)=f(t_F) = 0$ (so, are discontinuous at the boundaries - points of measure zero).

Specifically, I want to know the name of the space that tightly/exclusively contains these functions to look for references and their properties: real one-variable non-constant functions of the form: $$f(t) = x(t)\cdot(\theta(t-t_0)-\theta(t-t_F))$$ where $\theta(t)$ is the unitary step function, $t_0 < t_F$ (if required, assume is defined only for $t_0 \leq t \leq t_F$), $f(t_0)\neq 0$ or $f(t_F)\neq 0$ or both, and $x(t)$ is a "fully" continuous and smooth non-constant real-valued one-variable function $ x(t) \in C^\infty$.

An example of "bump function" $\in C_c^\infty(\mathbb{R})$: $$ f(t)= e^{1-\frac{1}{1-t^2}},\text{ }|t|\leq 1$$

Example of "time-limited bounded function" $\notin C_c^\infty(\mathbb{R})$: $$ f(t)= e^{1-\frac{1}{1-t^2}},\text{ }0\leq t\leq 1$$

In some places I have found some "issues" about how the space of Bump functions is defined:

In some places they say is the space of compact-supported smooth functions $C_c^\infty$, but in other places they also required that they have to fulfill the requirements to be a "Mollifier" (more restricted, since it require that the function must behave as a Dirac delta function in the limit $\lim_{\eta \to 0} \frac{f(\frac{t}{\eta})}{\eta^n}=\delta(t)$, and $\int_{\mathbb{R}^n} f(t) dt =1$, or even that $f(t) \geq 0, \forall t$ and $\max_t f(t) = 1$).

So I am going to explicitly differentiate the space of Bump functions $C_c^\infty$ from the space of "Mollifiers", in here, lets say its $C_0^\infty$.

PS: Please, if you can, create a TAG for "bump-functions", there a lot of related questions and I don't know where to "place them". Beforehand, thanks you very much.

Added later

I think I found a method that could help to make these kind of spaces of time-limited function by avoiding the discontinuity of the functions in the borders of the compact-support... please review it on my second answer to this question.

2nd Added Later

I have found the following paper named "Finite time differential equations" by V. T. Haimo (1985), where continuous time differential equations with finite-duration solutions are studied, an it is stated the following:

"One notices immediately that finite time differential equations cannot be Lipschitz at the origin. As all solutions reach zero in finite time, there is non-uniqueness of solutions through zero in backwards time. This, of course, violates the uniqueness condition for solutions of Lipschitz differential equations."

Since, linear differential equations have solutions that are unique, and finite-duration solutions aren´t, finite-duration phenomena models must be non-linear to show the required behavior (non meaning this, that every non-linear dynamic system support finite-duration solutions).

The paper also show which conditions must fulfill the non-linear differential equation to support finite-duration solutions, at least for first and second order scalar ODEs.

Since one-variable scalar finite-duration functions are example of one-variable compact-supported functions, I want to know if this issue related to the non-uniqueness of the solutions (which, if I have understood it right, it means that more than one answer could behave as the intended function in the specific compact-support, don´t knowing now how it could be possible, maybe behaving different outside the compact-support?), could be instantly discarding the possibility of building a space of finite-duration functions (or more widely speaking, of non-smooth compact-supported functions). Hope someone can comment about it.

Xander Henderson
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Joako
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    When you write $C^\infty_c,$ what is the domain space? Often it's $\mathbb{R}^n,$ but your example $f(t)= {\cos^2\left(t\cdot\frac{\pi}{2}\right)},\ |t|\leq 1$ is not in $C^\infty_c(\mathbb{R}).$ – md2perpe Oct 12 '21 at 18:58
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    @md2perpe I take that example from here and adjusted to be in the interval $[-1; 1]$, so, Could you explain why is not a bump function? (is actually what I am trying to understand) – Joako Oct 12 '21 at 23:24
  • Another indication that you'll need to refine your expectations: finite linear combinations of such functions can have finitely-many jump discontinuities... and infinite linear combinations could have infinitely-many. Do you really want/need such functions... or, for example, would you really prefer functions that are continuous except at "endpoints", etc.? There may be problems with making that a vector space... – paul garrett Oct 13 '21 at 00:24
  • @paulgarrett I want to understand the kind of functions that model physical phenomena: reading about Bump Functions, I realize that every continuous and differentiable signal that has a starting point, and endpoint, and bounded energy, can't be analytical, so every solution I have seen through standard functions are really approximations. I am self studying from Wikipedia and I found some issues/contradictions on the definitions so I am a bit lost. I read about Bump functions which are within $C_c^\infty$, but it required to begin and end at $0$, and this is too restrictive to model anything. – Joako Oct 13 '21 at 00:38
  • Where on that page does it say that a function like that is a bump function? – md2perpe Oct 13 '21 at 07:35
  • @md2perpe is a function defined in a close interval $|t| <1$ (compacted-supported I believe is named), bounded (since $\max = 1$ and $\min = 0$), is smooth a think since its derivatives will be always combinations of trigonometric functions, and in the edges is smooth since $\lim_{t \to \pm 1} f(t) = 0$ and $\lim_{t \to \pm 1} \frac{df(t)}{dt} = 0$. In the mentioned link they use in the comments the function $(1+\cos(x))^2$ as example of a bump function which has known Fourier transform in closed form... Can you explained why that was a bump function and why my example is not? – Joako Oct 13 '21 at 16:50
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    Okay, they say that it is compactly supported, but it is not in $C^\infty_c(\mathbb{R})$ as you write since $f^{(4)}$ is not continuous at $x=\pm\pi.$ – md2perpe Oct 13 '21 at 18:43
  • @paulgarrett I was thinking about the linear combinations you mention, but I believe if they are finitely many, it will be a similar problem that the defining of Cumulative Distributions functions through Cadlag functions, if they are finitely many jumps it going to be a definable thing since its Lebesgue measure its still going to be finite. – Joako Oct 13 '21 at 18:44
  • @md2perpe thanks, now I understand that for be a Bump function it also has to fulfill $\lim_{t \to \partial t} \frac{d^n f(t)}{dt^n} = 0$.. I will change the selected example to properly asked the question. – Joako Oct 13 '21 at 18:58
  • @paulgarrett I think I found a method that could help to make these kind of spaces of time-limited function by avoiding the discontinuity of the functions in the borders of the compact-support... please review it on my second answer to this question. – Joako Nov 10 '21 at 10:17

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