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As I had ever know there are at least two (previously unsolved) problems motivate the study/development of abstract algebra: (1) the ancient Greeks' three problems in compass-and-straightedge construction, and (2) the problem of solving polynomial equations of degree five or higher.

Although to me it is fascinating to study mathematicians' insights into these kind of problems, I haven't understand clearly what (previously and currently unsolved) problems motivate the study/development of analysis.

Could someone please explain like you are teaching a math undergraduate with some mathematical maturity? Thanks in advance!

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  • $\begingroup$ I think to be able to define differentiation is a good motivation already? $\endgroup$
    – user99914
    Commented Oct 4, 2015 at 11:04
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    $\begingroup$ One motivation is just to prove everything in calculus very carefully. Maybe another motivation is to understand Fourier series clearly -- like, which functions can be represented (and in what sense) by a Fourier series? I think Stein's books on analysis use Fourier series as motivation for a lot of the material. $\endgroup$
    – littleO
    Commented Oct 4, 2015 at 11:06
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    $\begingroup$ topics such as integration (with lebesgue integration as generalisation), differentiation, continuous but nowhere differentiable functions, existence of solutions to pde should give enough motivation for studying analysis $\endgroup$
    – u184
    Commented Oct 4, 2015 at 12:11

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The essence of calculus is neither in the proofs nor in the exercises, and famous particular problems, like the brachystochrone, are more of an anecdotic nature. The main issue is that calculus (like other fields of mathematics, e.g., probability theory) provides a zoo of concepts that allow to formulate and better understand the laws of nature by setting up precise paradigmatic, or "toy", models of intricate situations.

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The answer really depends on what will be covered in your real analysis class, and this can change depending on location and professor. One motivation is a rigorous construction of the real numbers. Another is proving and using more theoretical theorems and notions from Calculus (when can I pass a derivative inside an integral?). As already mentioned, Fourier Series provide excellent motivation, and the Stein and Shakarchi texts are really excellent in this regard. They also explain how to use Fourier Series to solve various PDEs, and aside from separation of variables this is likely the only other solution method many undergraduates learn about solving PDEs.

The Stein and Shakarchi texts (taken as a whole) cover much more than a typical undergraduate real analysis course, however. The general theme of real analysis at the undergraduate level is "grappling with infinity" (taken from the excellent set of video lectures from Harvey Mudd).

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