Why study different kinds of functions in Real and Complex Analysis?

[Sigma057]

I've taken basic undergraduate Real and Complex Analysis, and I've noticed they focus on different kinds of functions. Real analysis studies things like Dirichlet and Cantor functions with infinitely many discontinuities while complex analysis studies mostly differentiable functions.

My question is, why don't we study both continuous and discontinuous functions equally in both disciplines?

I realize that this question may disappear at graduate level, but my question is directed at undergraduate-level material.

 

[AlephZero]

Here's an analogy. If you try to study polynomials using nothing but real variables, you soon get tangled up by the fact that a polynomial of degree n with real coefficients can have any number of real roots from 0 to n. If you want to get some general results, you need to consider complex roots. Then, you might as well consider complex coefficients as well, since restricting the coefficients to real numbers doesn't gain you anything very interesting.

The same is true for studying analytic functions, which is one of the main topics in complex analysis. You can't get very far studying the convergence of power series, for example, if you restrict yourself to real numbers. The useful concept is convergence in a circular region of the complex plane, not convergence on an interval of the real line which just happens to intersect the circle.

On the other hand, all the messy stuff that can happen with discontinuous real functions doesn't change much for complex functions, so you if you study it in Real Analysis there is not much educational value in studying it again in Complex Analysis.

 

[micromass]

^Right. The only truly different concept in complex analysis is that of complex differentiability, the rest is exactly the same as in real analysis. And it happens that a complex differentiably function is extremely well-behaved, so we're forced to study well-behaved things.

 

[glance]

Can I drop a very related, but slightly more specific question? (it seemed to not worth a new topic on its own, but if you do not think so I will delete this post and go for that option)

I get the importance of holomorphic functions, and the relative uselessness of studying continuous but not holomorphic complex functions. However, what strikes me is that this goes so far that even the concept of a singularity in complex analysis is defined only for holomorphic functions (referring for example to the wikipedia article on the subject). Is there some particular reason for this, other than the fact that the study of singularities of "simply continuous" complex functions may not worth the effort?

 

[WWGD]

How about analytic continuation?

 

[FactChecker]

I've taken basic undergraduate Real and Complex Analysis, and I've noticed they focus on different kinds of functions. Real analysis studies things like Dirichlet and Cantor functions with infinitely many discontinuities while complex analysis studies mostly differentiable functions.

The Dirichlet and Cantor functions are examples of bizarre behavior that are presented in the simplest context possible. Once the examples are given in a simple setting there is no reason to discuss them again in a more complicated setting.

My question is, why don't we study both continuous and discontinuous functions equally in both disciplines?

Discontinuous and non-differentiable functions are studied in multivariate calculus. There is no reason to consider those spaces as complex planes for those functions because the beautiful results of complex analysis do not hold.