0
votes
1answer
28 views

What does it mean “sequence with infinite range”

I'm trying to understand this phrase Find a sequence with infinite range that converges only to $0$. What does it mean "sequence with infinite range"? Thanks
0
votes
0answers
25 views

Terminology: 'inverse' of non-strictly monotonic function?

Suppose I have a nonincreasing function, $x \mapsto f(x) = y$. I want to call the function $$y \mapsto g(y) \triangleq \sup\{x:f(x) \geq y\} $$ the `inverse' of $\ f$ for brevity, despite that $\ f$ ...
0
votes
0answers
16 views

How to call a function defined on a set with gaps on arbitrarily small scales.

Let $I$ be an interval and $A\subset I$ such that for any two points $x,x'\in A$ there exists an interval $J$ between $x$ and $x'$ such that $J\cap A=\emptyset$. How does one call this proerty of ...
0
votes
1answer
29 views

Terminology: Alternatives for zero crossing

Is it correct to name the red and blue points hinge points, as an alternative to zero crossing? Or are their better terms to describe these points? Update I have several functions like these. I ...
0
votes
1answer
16 views

Difference between $C_0^{\infty}(U)$ with support in $A$ and $C_0^{\infty}(A)$

Let $A \subseteq U$ be open sets of $\mathbb R^n$. Is it true that $$ \lbrace f \in C_0^{\infty}(A) \rbrace = \lbrace f \in C_0^{\infty}(U) : \text{support of } f \subseteq A \rbrace \quad ? $$ I ...
3
votes
1answer
45 views

Local topological properties

Could we define the terms locally connected/compact/contractible/simply-connected/whatever to mean that there is a basis (for the topology on our space) of ...
1
vote
2answers
75 views

Using sequential definition of functional limits, show that $\lim_{x \rightarrow 0} 1/x$ does not exist

Using sequential definition of functional limits, show that $\lim_{x \rightarrow 0} 1/x$ does not exist I have two questions regarding this. Firstly, say we have a function that 'converges' to ...
14
votes
2answers
721 views

Why is analysis called “analysis”?

Just as the topic says, how did the name "analysis" come to denote the specific mathematical branch dealing with limits and stuff? The term "analysis" seems very generic compared to the words for the ...
1
vote
1answer
30 views

What is the name of this measure property?

if we have a function $f \in L^p$ sucht that $||f||_p =1$ and $m$ being a finite measure. Define a new measure $\mu$ by $$\mu(A):=\int_A |f(x)|^p dm(x).$$ Then $\forall \epsilon > 0 \ \ ...
0
votes
0answers
21 views

Definition of Range as Minimal Interval Containing Codomain

I am studying continuous functions where the domain is some interval (which may or may not be bounded, closed, etc). I am thinking about how continuity is related to other function properties, ...
0
votes
1answer
105 views

What's the name of $x^x$?

I know that $$f{(x)} = a^x$$ is called exponential function and $$f{(x)} = x^a $$ is the power function. But what is the name of $f{(x)} = x^x$?
5
votes
2answers
195 views

Is $(-\infty,\infty)$ a closed **interval**?

Note that we are working in the reals, not the extended reals. Do you understand a closed interval as "an interval that is a closed set" or as "an interval that includes both its endpoints"? If the ...
1
vote
1answer
279 views

What is a tail sequence?

The question is self-explanatory. What is a tail sequence or a tail of convergent sequence? Thanks
2
votes
1answer
193 views

Bounded open interval

This is just a quick terminology question. My textbook is talking about the continuity of $ f $ over a bounded open interval $ (a,b) $. Am I right in assuming that this means $a$ and $b$ are finite? ...
3
votes
2answers
129 views

Alternatives to percentage for measuring relative difference?

If the value of something changes from $\;a\;$ to $\;b\;$, their relative difference can be expressed as a percentage: $$ \newcommand{\upto}{\mathop\nearrow} (D0) \;\;\; a \upto b \;=\; (b-a)/a ...
0
votes
1answer
179 views

What is an “essentially sharp” estimate?

I frequently encountered theorems here and there, which claim to establish essentially sharp upper bounds for certain quantities. However, I am confused about what “essentially sharp” exactly means, ...
1
vote
1answer
29 views

What is a Linear Set of Points.

According to the classroom notes "Uniformly Continuous Linear Set" in American Mathematical Monthly, Vol. 62. No. 8(Oct., 1955) pp. 579-580, Author: Norman Levine He let E denotes a Linear Set of ...
1
vote
2answers
93 views

Replacing the value of a function with the value of the limit - is this a standard construction?

Consider a partial function $f : X \rightarrow Y$ where $X$ and $Y$ are topological spaces and $Y$ is Hausdorff. Note that, although the source of $f$ is $X$, the actual domain of $f$ is a (not ...
0
votes
1answer
36 views

Terminology of functions in $L_2$

I am reading a text that states ... any function in $L_2(0,\pi)$ has a Fourier sine series that converges to it in $L_2(0,\pi)$ ... Unfortunately no definition of $L_2$ is given. What does ...
1
vote
1answer
27 views

What is the name for functions with this property similar to concavity?

What's the name of the class of functions satisfying the following property: Given some convex space $D\subseteq \mathbb{R}^k$, a function $f:D \to \mathbb{R}$ is called $\mathbf{(?)}$ if for all ...
2
votes
1answer
61 views

What does this phrase about the weak topology of bounded operators mean?

Can somenone remind me of the meaning of the following statement: the family of operator valued functions $A(\omega)$ converges to $A(\omega ')$ in the weak topology of bounded operators from ...
1
vote
2answers
34 views

I'm having trouble with math terminology, when is it appropriate to say “mapping” or “maps to”?

For example, in a homework assignment I need to prove that a metric space $M$ is isometric to itself. If I say that $M$ maps to $M$ is that the same as saying that theres a function that takes ...
1
vote
1answer
61 views

What is meant by the word ‘control’ in the context of analysis?

Wikipedia’s article on Dini’s Theorem states: This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by ...
1
vote
1answer
437 views

Defintion of Upper & Lower Riemann Sum

I recently came across the terms: 'upper Riemann sum' and 'lower riemann sum'. Are they represent the same things as of 'upper sum' and 'lower sum' defined as follows.
0
votes
1answer
72 views

What does $f$ is measurable as a function $f^{-1}(\mathbb{R})\to\mathbb{R}$ mean?

I saw a question where we have $\overline{\mathbb{R}}:=\mathbb{R}\cup\{\pm\infty\}$ and $(X,S)$ is a measurable space, $f:\, X\to\overline{\mathbb{R}}$ In one part I was told to assume that $f$ is ...
0
votes
1answer
113 views

What does it mean for a set to be “nested”?

What does it mean for a set to be "nested"? and can you please show an example of that is and one that isn't
1
vote
1answer
196 views

Name for the real numbers between $0$ and $1$

I see this class of numbers all the time, so I was wondering if there was a special name for it. How to refer to a number $n$ in $\Bbb R$, such that $0<n<1$?
2
votes
0answers
153 views

Property similar to subadditivity

A function is called subadditive such that $f(x+y)\le f(x)+f(y)$ holds for any $x$, $y$ in the domain of $f$. (Let us say that, for example, the domain is some subset of $\mathbb R$ closed under ...
1
vote
1answer
247 views

Is there a different name for strongly Darboux functions?

A function $f\colon\mathbb R\to\mathbb R$ is called Darboux function, function with Darboux property or function with intermediate value property for for any $a<b$ and any $z$ between $f(a)$ and ...
3
votes
1answer
1k views

Cluster point of a function at a point

This post is quite long, since I wanted to include the necessary context. (Maybe I've put there too much.) Maybe you might prefer to look at the questions at the end of this post, first. ...
1
vote
3answers
2k views

why do we use 'non-increasing' instead of decreasing?

In english based math language it seems that non-increasing $\Longleftrightarrow$ less or equal (non-strict decreasing) decreasing $\Longleftrightarrow$ strict less ( strict decreasing) ...
2
votes
0answers
303 views

Are derivative and differential the same thing?

(Sorry for bad English.) Let $f:\mathbb R^n\to\mathbb R^m$, $x\in\mathbb R^n$. What is a drivative $f'(x)$? It is the linear map $f'(x):\mathbb R^n\to\mathbb R^m$, $h\mapsto ...
3
votes
3answers
263 views

Is there a name encompassing both limit inferior and limit superior

Is there a mathematical term which would include both liminf and limsup? (In a similar way we talk about extrema to describe both maxima and minima?) The only thing I was able to find was that some ...
8
votes
1answer
192 views

What kind of completeness is the completeness of $\mathbb{R}$?

As opposed to the algebraic completeness of $\mathbb{Q}$, which yields the algebraic numbers, we can say that $\mathbb{R}$ is complete in the sense that every non-empty subset of $\mathbb{R}$ bounded ...
18
votes
7answers
6k views

Why does “convex function” mean “concave *up*”?

A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line ...