# How to explain the concepts of limits and continuity to non-mathematical students

How can I explain the fundamental concepts of limits and continuity to a student with a non-mathematical background?

I am a PhD student in Mathematics working in Differential Geometry. As a part of my teaching assignment I need to explain the concepts of Limits and Continuity to some undergraduates majoring in Physics.

I don't know how to explain these concepts easily to them. Will anyone at all listen to me?

Should I make a slide presentation to them? Is it possible to explain mathematical concepts using PowerPoint/LaTeX?

Are there any sites or online resources which can help me to make them understand Mathematics in an easier way?

• Get a good book. Follow it closely. They have likely spent more time considering this, testing, and refining than you have available. (Suggest OpenStax Calculus if none other available.) Mar 16, 2018 at 12:30
• You may be interested in reading matheducators.stackexchange.com/questions/13777/…. In no case try to teach the wrong idea that the limit is the state after infinitely many terms of the sequence. Mar 16, 2018 at 12:49
• Slide presentations have turned out useful in my classes. The following slides on the limit are in German but should show the essentials. When looking at them please apply the animated screen representation of PowerPoint hs-augsburg.de/~mueckenh/Transfinity/M19.PPT Mar 16, 2018 at 13:06
• Don't confuse format with content. Whether the material is explained in a slide or formatted with LaTeX is probably much less important than what you say. Mar 16, 2018 at 21:00
• How do you define a "non-mathematical" background? Undergraduates majoring in physics have more mathematical background than many,
– J W
Oct 1, 2022 at 7:30

My advice is to pick one or the other, not both, for topic, probably continuity.

Also, physics students are not a-mathematical. They may not be PhD geometrists. But they are definitely people that cleave to the standard algebra and calculus track.

I would avoid the trap of thinking of them as unwashed non Rudinites that you are going to correct gaps of. Instead, frame your talk more to explore the different ways one can think about continuity, or perhaps with a msth history flair, discuss how and who had different frames over time. On other words more like a fun pop video on YouTube.

It is fine to mention some very advanced concepts without feeling you need to teach and prove. I get value out of knowing what a topic is, who works on it, what classes cover it, without knowing the content itself. Like having a situational awareness of what an abstract algebra course us, without having ever done one. Similar to how you (should) have a general awareness of what an organic chemistry course is, without having done one. But more as a generally educated person to know whatvthe fields of knowledge are.

Format is separate from the aspect of how you slant the talk. But my advice is to do a chalk talk. And even for that, I would do more of an entertaining speech, not a step by step derivation. Ie less chalk than a teacher. Just occasionally as needed, for some example or diagram.

Don't do slides. Will put them to sleep. Death by PowerPoint. And it will make it to easy for you to fall into the trap of being too hard and too proofy.

• non Rudinites that you are going to correct gaps of --- Probably your use of "gaps" here was the common/natural language usage, and thus the word play I immediately noticed was not intended (i.e. like "pun not intended", said when it really wasn't intended), but the word "gap" is a commonly used technical term in the construction of real numbers, a construction that Rudin is an often cited text reference for. Thus, it somewhat makes sense to say "correct the gaps", although one usually says "fill in the gaps". Oct 2, 2022 at 14:29

My former adviser explained continuity in this way to a freshman class (all math majors):

Assume that you are taking shower. The independent variable is the position of the shower handle and the dependent variable is the water temperature. If rotating the handle by small amount, you change the temperature just a little, you have continuity but if the slightest touch applied to the handle changes the temperature from icy cold to boiling hot, then you have a discontinuity. And you have, probably, seen both kinds of showers, haven't you?

Instead of the shower, you can put a ball on a flat surface and then on a convex one (a piece of wood, a sheet of transparent plastic, some old bearing, a few screws and 10 minutes of work, and you have the device you need). Then tilt the surface a bit. On the convex one, the position of the ball will change only a bit with rotation, but on the flat one it will travel from one end to another when you cross the horizontal position. Same idea, of course, but slightly less wet and messy.

I should say that I cannot reproduce how he proved most of the theorems today, but 38 years later I still remember the shower story. So, perhaps, your students will remember it too. And it can easily be converted into the formal $$\varepsilon-\delta$$ statement, should you choose to go that far.

Just my two cents.