# Questions tagged [definitions]

For questions related to the issue of concepts of definitions.

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### Why is continuity defined as a local property?

The formal definition of continuity is a local property (the definition of continuity at a point is a property of the germ of the function at this point). Why is it a good decision to make the ...
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### How can I motivate the formal definition of continuity?

In order to teach continuity of real valued functions $f:D\to\mathbb R$ one may start with the (in some sense wrong) intuition $f$ is continuous when its graph can be drawn without lifting the pen. ...
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### Defining vertical tangent lines

In looking at the definition of vertical tangent lines in some popular calculus texts, I noticed that there are a few different definitions for this term, including the following: A function $f$ ...
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### What is the intuition behind the limit superior?

I want to write an article which explains the limit superior. I also want to present the intuition behind this concept. Currently I would describe the limit superior as the "least upper bound of a ...
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### Rigorously defining the concept of an angle for high school students

Arriving at a rigorous definition of the concept of angle for high school students is not as easy as expected. Google search provided me with many definition that are too technical or too vague IMO. ...
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### Can students tell the difference between the "definition if" and the "theorem if"?

The word "if" is used in two meanings in mathematics: Definition. A topological space is compact if every open cover has a finite subcover. Theorem. A topological space is compact if it is ...
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### What are the questions one should ask of oneself in trying to understand definitions and lemmas? [closed]

Definition (Finite series). Let $m,n$ be integers, and let $(a_i)_{i=m}^{n}$ be a finite sequence of real numbers, assigning a real nmber $a_i$ to each integer $i$ between $m$ and $n$ inclusive. Then ...
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In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [$a(b + ... • 21.3k 8 votes 2 answers 310 views ### Is multiplication by zero clear for and understood by K-3 students? For K-3 students, perhaps it is not acceptable to introduce multiplication by zero as a property or definition. Instead, the child may think about multiplication as, e.g., repeated addition. Examples ... • 1,127 4 votes 2 answers 2k views ### how is volume different than capacity I have in mind that volume is the amount of room or space a 3-d object takes up - its "outsideness" and that capacity with the amount of room or space a 3-d object can hold. Then I start thinking ... • 51 10 votes 2 answers 298 views ### What is a good way to explain the slightly different kinds of continuity? What is a good way to explain the slightly different kinds of continuity to students? I have in mind these kinds of continuity: A function is continuous at a point. (This also has two sub-kinds: ... • 2,504 14 votes 4 answers 427 views ### Explaining subjects whose justification requires demanding technical content This is my first question and I hope it's appropriate. Often in the process of teaching a subject I start with examples of a phenomenon, exhibiting similar properties between the examples and ... • 365 20 votes 9 answers 3k views ### The definition of natural log and e I'm asking this question from the point of view of an introductory non-rigorous calculus instructor. Calculus textbooks have different approaches about how to define$e$and$\ln$. For example, my ... • 19.1k 21 votes 8 answers 4k views ### Are there disadvantages to teaching complex numbers as purely geometrical objects? Complex numbers are, or at least were to me, generally introduced like this: There's no number whose square is negative. That's a shame! Well, whatever - we'll make one up! Set$i^2=-1$and declare ... • 1,277 20 votes 9 answers 2k views ### Why do students have problems with showing that something is well-defined? How can this be improved? I see a lot of students struggling when they have to show that something is well-defined. I have the feeling that this is often not understood. Two examples: When defining a sequence$x_n= g(x_{n-1}...
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For some objects there are alternate definitions, which are "morphism-oriented". To give some examples, there are two definitions of a prime number: $p$ is prime if it is greater than $1$ and has no ...