# What is the most fundamental continuous function in calculus after a constant (totally straight) line?

What would be to teach in most countries and education systems?

(Taught before "high education frames", i.e. before doing bachelor of arts in mathematics).

• I think that this question might profit from additional information. Without context I am not sure whether I fully understand what kind of answer would be helpful for you. Oct 25, 2021 at 13:48

The linear function is a pretty good first example: they are very fundamental and everything is simple. But rather than think in terms of the most fundamental thing to come next, I would encourage you to think in terms of typical examples that show the breadth of the concept.

• Linear function
• Polynomial of maybe second degree (not linear, but still fairly familiar)
• Constant function (too elementary a case for students to understand easily, and valuable due to that)
• Absolute value (or some other piecewise defined function)
• Square root (not defined everywhere)
• Maybe $$x^{-1}$$ (if you want to get tricky)

For signal processing you might go more in the direction of sine and cosine as simple examples and in general use more periodic examples.

• With all niceness I did not ask for any encouragement, just for an example (and it's not even about mere mathematics but about signal processing); I had a mistake asking for just one example because I would agree that constant function (100% straight line) is too simple and I need the thing coming just above it in the line so there should be just two examples. I should edit the question to explain and I think it would be good if you will edit the answer accordingly. Oct 19, 2021 at 5:48
• I also disagree with Constant function (too simple a case, causes problems) because I believe that this function should be the very first one to teach when introducing the "continuous" concept (at least as taught for signal processing). Oct 19, 2021 at 5:51
• Edited. You might want to add something about signal processing into the question. Oct 19, 2021 at 7:11
• The most elementary is the most elementary (as you know). I think that a constant line, and then sine, are enough for giving an elementary firstmost glimpse into signal processing (when explaining a term such as "continuous"). Oct 19, 2021 at 8:01
• @chichorozov Since you seem to already know the answer you want to get, you should answer the question yourself. It is okay to do. Oct 19, 2021 at 9:06

The category-theorist in me would argue the most fundamental function is the identity function, so $$f(x)=x$$ might be a good one to go to next.

I say this one because, when you really start diving down into what "fundamental" is, the existence of an identity mapping is really key to a lot of structures. In particular, categories. And, of course, its a dirt simple function, which is probably what you're going for.

Of course this being said, the constant function and identity function can also be tricky first functions because they are so simple that it's hard to get any feel for functions at all with them.