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In the Apostol's Calculus (at first) integrals are treated without limits but derivatives are postponed to come after limits, their conceptual foundation.

I want to know that could the author have defined derivatives similar to the he defined integrals? If yes can you write how? If not, what is the conceptual difference between derivatives and integral that makes this happen?

edit: (My question is different from the general question of defining derivatives in terms of other things as it demands a similar way the author defines integrals.)

EDIT2:The author uses the darboux integral,Which is equivalent to the riemann integral;Now it now remains to be seen if there is a way to define a darboux-esque derivative too?

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  • $\begingroup$ @caverac my question is limited to the way the author of this book defines it and If possible I want a way simliar to his for derivatives, not just any way to make my thinking consistent $\endgroup$ – Logan Luther Nov 27 '17 at 11:13
  • $\begingroup$ but the Darboux integral is defined through limits, the limit of a Darboux upper or lower sum. $\endgroup$ – Masacroso Nov 27 '17 at 11:32
  • $\begingroup$ @mascroso in the book it is defined as sup's and inf's and those are taken from the completeness axiom of real numbers. $\endgroup$ – Logan Luther Nov 27 '17 at 11:51
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See the following two papers, which I found rather quickly (less than 30 seconds) by googling:

[1] Oved Shisha, Derivative without limit, Journal of Mathematical Analysis and Applications 113 #1 (January 1986), 280-287.

[2] Mordechai Falkowitz, Oved Shisha, and Nehad N. Morsi, Note on derivative without limit, Journal of Mathematical Analysis and Applications 127 #2 (1 November 1987), 595-597.

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  • $\begingroup$ the first text seems based, in the hood, in the concept of limit. It is using $\epsilon-\delta$ reasoning. $\endgroup$ – Masacroso Nov 27 '17 at 11:28
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    $\begingroup$ @Masacroso: No $\epsilon-\delta$'s appear in the statement of Theorem 2 or the definition of "direction". However, the fact that "direction" is defined by looking at a property true in all neighborhoods of a point seems half-way cheating to me, but only half-way because for example a function can be strictly increasing at a point and fail to be strictly increasing in any neighborhood of the point (i.e. holding in all sufficiently small neighborhoods isn't always the same as holding at the limit). $\endgroup$ – Dave L. Renfro Nov 27 '17 at 11:38
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    $\begingroup$ @Masacroso: Actually, my example in the last comment is slightly off the mark, because holding "at a point" and holding "in a limit taken to the point" are not always the same thing, and there are actually (at least) three different notions that can sometimes be defined for certain properties: (1) holding in every neighborhood of the point; (2) holding in a limit taken to the point; (3) holding at the point. And with this, I need to leave, as there is somewhere I need to go (offline). $\endgroup$ – Dave L. Renfro Nov 27 '17 at 11:50
  • $\begingroup$ I will take a better look to the articles. $\endgroup$ – Masacroso Nov 27 '17 at 11:52
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The problem with any other definition of the derivative of a function is that it is hiding it limit nature. One can says, as someone said in other answer, that the derivative is the slope of the "tangent" at a point or that it is "the best linear approximation" to the function.

But all these notions rely, under the hood, in the concept of limit, so a derivative cannot be defined rigorously without the use of limits. The same for integrals.

So any definition of derivative that doesnt use limits (or some notion that include the concept of limit, by example supremum, nets and other set-theoretic constructs) cannot be said that are rigorous or formal definitions, more like informal or intuitive definitions.

In short: you can define in mathematics things in many ways, but not all of these definitions can be considered formal or rigorous.

EDIT: my point of view is similar to this one, relative to the difference between intuitive notion and formal definition.

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  • $\begingroup$ please note my latest edit. $\endgroup$ – Logan Luther Nov 27 '17 at 11:31
  • $\begingroup$ @Masacroso that's exactly also my point of view, if we are talking of calculus yhe derivative concept passes by yhe limit concept otherwise we are talking about something else $\endgroup$ – user Nov 27 '17 at 12:44
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    $\begingroup$ Maybe you can see the post: axiomatic-approach-to-differential-calculus and try to follow the path to: Kock–Lawvere axiom and Synthetic Differential Geometry. $\endgroup$ – Mauro ALLEGRANZA Nov 27 '17 at 13:21
  • $\begingroup$ @MauroALLEGRANZA thank you very much. I will take a look at these articles. $\endgroup$ – Masacroso Nov 27 '17 at 13:39
  • $\begingroup$ infact that's no longer a calculus but differential algebra another or Others stuff wich belong to different mathematical area, it's obvious you can define derivative without limit if you assume derivative as an axiom $\endgroup$ – user Nov 27 '17 at 13:48
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Don't know about the book/author you're talking about. However you could define the derivative at one point as the best linear approximation of the function. This could certainly be done and will be equivalent to the typical definition of the derivative. Not sure however how much could be said about derivatives without ever using limits. Maybe you would have a definition, but coultn't use it for much.

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