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.


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.

  • $\begingroup$ 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? $\endgroup$ Oct 18, 2021 at 14:36
  • $\begingroup$ by the way, for all practical purposes, the definition of "continuous signal" is identical to that of "continuous function". Do you know the latter? $\endgroup$ Oct 18, 2021 at 14:53
  • 1
    $\begingroup$ Yup, it'd have to be a child who understands Calculus. $\endgroup$
    – TimWescott
    Oct 18, 2021 at 15:25
  • $\begingroup$ Do you mean continuous in time, or do you mean a continuous function of time? $\endgroup$
    – TimWescott
    Oct 18, 2021 at 15:25
  • $\begingroup$ @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". $\endgroup$
    – TimWescott
    Oct 18, 2021 at 15:26

1 Answer 1


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.

  • 1
    $\begingroup$ The mischievous part in me tells me to add: "you could draw a sampled signal (staircase) that way, too". But that's a nice way of explaining for someone young. $\endgroup$ Oct 19, 2021 at 6:42
  • $\begingroup$ Sure, one can roll the cone of the pencil without marking the paper, draw on a folded sheet of paper. Then it becomes interesting $\endgroup$ Oct 19, 2021 at 6:54
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    $\begingroup$ We'd better stop before we get to fractals. :-) $\endgroup$ Oct 19, 2021 at 11:41
  • $\begingroup$ @aconcernedcitizen but that IS a continuous signal! $\endgroup$
    – user253751
    Oct 19, 2021 at 12:03
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    $\begingroup$ I have added an example of a classical wiggling function $\endgroup$ Oct 19, 2021 at 16:26

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