Questions tagged [definitions]

For questions related to the issue of concepts of definitions.

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18
votes
7answers
863 views

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. ...
17
votes
5answers
599 views

What is a variable?

There are two kinds of answers I'm looking for: What do students think a variable is? What do YOU, the teacher, think a variable is? I'm also interested in why you think a variable is what you think ...
19
votes
9answers
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}...
19
votes
9answers
2k 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 ...
4
votes
2answers
143 views

Does solving crosswords help with recall of definitions?

I have an idea to help with self-study and recall of definitions. It is to create a crossword where the clues are the definitions and the words to solve are the concepts being defined. I haven't ...
4
votes
2answers
511 views

Why is continuity only defined on its domain?

As mentioned in this question students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \to \...
3
votes
1answer
474 views

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 ...