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Please share an answer with a simple example from the "daily" (colloquial) lives of humans for a signal which is "continuous" and explain what is it that rigorously makes it "continuous".

Please use the simplest language you can, as if you would explain it to a child.

Update

Before creating this post I already done some research and found this definition:

A continuous time signal is a function that is continuous, meaning there are no breaks in the signal.

But I thought that there could be an explanation which is not "mathematical" (no "time" and "function" as applied in mathematics).

Documenting a discussion in comments

um, "explain without math": what purpose does that serve? What do you need to be able to do with that knowledge afterwards? Because, if you don't know what a function is, well, honestly, thinking about continuous vs non-continuous signals makes little to no sense; you couldn't apply that knowledge anywhere.

I know what a function is.

Three possible purposes:

  • It will give a glimpse about with what signal processing experts work (curiosity seed).

  • It will allow a person to better categorize phenonmenons in reality

  • It will allow a person to understand a broader topic in which the term was reminded in a conversation.

I will also add that in Mathematics education it's good to start learning elementary concepts by daily life ("colloquial") examples and if this matter of "signal processing" is nonetheless a mathematical one than an example from daily life might raise the chance a student would learn "abstract" data about that.

Alkalix
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  • You can't "rigorously" define continuous to a child. That doesn't work. The rigorous definition is math at a level that you wouldn't call child-appropriate. (Some children nonwithstanding). What do you really want, a child-appropriate, or a rigorous explanation? – Marcus Müller Oct 18 '21 at 14:36
  • by the way, for all practical purposes, the definition of "continuous signal" is identical to that of "continuous function". Do you know the latter? – Marcus Müller Oct 18 '21 at 14:53
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    Yup, it'd have to be a child who understands Calculus. – TimWescott Oct 18 '21 at 15:25
  • Do you mean continuous in time, or do you mean a continuous function of time? – TimWescott Oct 18 '21 at 15:25
  • @MarcusMüller: The ELI5 explanation could just be "it doesn't jump from one value to another -- there's always values in between, if you look close enough". – TimWescott Oct 18 '21 at 15:26
  • Tell him to look at a bee flying, then the same but blinking. Continue in these lines: apparently continuous and apparently sampled. – a concerned citizen Oct 18 '21 at 15:29
  • @TimWescott yup, that be the ELI5 explanation, but not rigorous. How do look closely enough? You'd quickly be introducing concepts like "for every neighborhood..." or "For every convergent series...", and that's probably the point where the five-year-old would start crying. – Marcus Müller Oct 18 '21 at 15:34
  • by the way, we don't "paste" answers. The answers here are written by experts in their spare time. Things you could easily find online yourself, we don't copy and paste – we'd just comment like "look here, and please next time do a little better research yourself (-1)" :) – Marcus Müller Oct 18 '21 at 15:36
  • Generally, you seem to have a background, so I wonder what about the definition you don't understand. I think you underestimate the hardship that is explaining math to a child and overestimate how hard it would be for you to explain exactly what you've learned so far about this and ask a precise question! – Marcus Müller Oct 18 '21 at 15:40
  • @MarcusMüller I want a primal, simplistic explanation, suitable for a child or for anyone totally new to the subject with elementary-basic level of mathematics; I am not sure if I know the latter (continuous function) but a simple example would help; I wrote "paste" by mistake (I confused). I already did some research myself before pasting the question and found this: A continuous time signal is a function that is continuous, meaning there are no breaks in the signal. – Alkalix Oct 19 '21 at 03:28
  • Okay I have edited to improve the post. – Alkalix Oct 19 '21 at 03:32
  • um, "explain without math": what purpose does that serve? What do you need to be able to do with that knowledge afterwards? Because, if you don't know what a function is, well, honestly, thinking about continuous vs non-continuous signals makes little to no sense; you couldn't apply that knowledge anywhere. – Marcus Müller Oct 19 '21 at 08:21
  • @MarcusMüller Just three possible purposes: It will give a glimpse about with what signal processing experts work. It will allow a person to better categorize phenonmenons in reality. It might also allow a person to understand a broader topic in which the term was reminded in a conversation. – Alkalix Oct 19 '21 at 08:34
  • @MarcusMüller I know what a function is. – Alkalix Oct 19 '21 at 08:34
  • @MarcusMüller if it's possible to be described without math, "colloquially" you are welcome to example it as a phenomenon from the life of basically any person on earth; if it's not and it's only a matter of math, you are welcome to example that too; any discussion would just waste time to us all (no pun intended). – Alkalix Oct 19 '21 at 08:36
  • no, this isn't a discussion forum, honestly. – Marcus Müller Oct 19 '21 at 11:43
  • @MarcusMüller of course it isn't, that's why I wrote Please share an answer with a simple example from the "daily" (colloquial) lives of humans for a signal which is "continuous" and explain what is it that rigorously makes it "continuous".. And Laurent Duval gave such an example. – Alkalix Oct 19 '21 at 11:52

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A standard way of describing a continuous curve before rigorous and provable defintions was:

What you can draw on a sheet of paper without lifting the pen

This is quite visual, and applies to signals as well. Of course this applies well to curves of finite length. Fractals and pathological functions like $x\mapsto x \sin 1/x$ (below) are examples of a need for more formal continuity axioms:

x sin 1 /x

If one wants to elaborate, one may start to discuss on what happens when you change the width of the tip of the pen, pencil or brush. This concept is close to different size of open sets.

calligraphy cursive letters

There are real example from calligraphy, especially cursive writing, or (sequences) letters you can draw in one stroke. This gets complicated as a bunch of letters have loops (hence are more curves or multivalued functions). But if this triggers the curiosity of the student, the game is won.

Laurent Duval
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