>> and some good experience manipulating continuous functions and their derivatives.

> Nope

1. "good experience manipulating differentiable functions and their derivatives" sounds weird in prose.

2. Some continuous functions are differentiable. Those ones have derivatives you can manipulate. In fact knowing when a function is not differentiable is a pretty useful skill.

> The interview was in their computer group, but I indicated that I'd like to get into quantitative trading -- they acted like they had never heard of any such thing.

Sounds like you interviewed for position X and talked about wanting position Y, and were rightly rejected as "not a good fit; likely to leave at first opportunity".

> only a very tiny fraction of the people on Wall Street had good backgrounds in measure theory and stochastic processes based on measure theory

Probably true. Think of this as "calculus for engineers vs. analysis", and imagine how well a civil engineering interview would go if you talked about different types of integrals instead of talking about how to use the basic stuff to build good bridges. Fact is most people in industry are looking for "calculus for engineers" levels of formal understanding. Enough to be useful and make money while avoiding expensive mistakes.

> and were looking to hire people like themselves.

Also probably true. This is a good assumption not just on Wall St but everywhere.

> Think of this as "calculus for engineers vs. analysis", and imagine how well a civil engineering interview would go if you talked about different types of integrals instead of talking about how to use the basic stuff to build good bridges.

That analogy shouldn't apply to quantitative trading on Wall Street: That challenge needs more than just engineering math approaches if only to read the literature.

E.g., apparently broadly the first cut way to evaluate exotic options is to use the Brownian motion solution to the Dirichlet problem, that is, the subject of Markov processes and potential theory. The subject is awash in measure theory, e.g., stopping times, the strong Markov property, regular conditional probabilities, of course conditioning and the Radon-Nikodym theorem. This isn't advanced calculus for engineers. E.g., the work of Marco Avellaneda at NYU Courant, Steve Shreve at CMU, no doubt the work of E. Cinlar at Princeton.

> Sounds like you interviewed for position X and talked about wanting position Y, and were rightly rejected as "not a good fit; likely to leave at first opportunity".

My point is not that I was rejected but just that they had no hint that anyone at Morgan Stanley was doing applied math for automatic trading.

If you do study (continuous) mathematics, one of the things you build up is a little stable of strange functions to help you test your intuition.

A very common "right of passage" exercise in introductory analysis is to come up with the Weierstrass function or something similar (usually with a little coaching). This is an everywhere continuous and nowhere differentiable function that then goes into your little toolkit.

Ah, a favorite is a function that is differentiable but its derivative is not Riemann integrable!

Recall, a function is Riemann integrable if and only if it is continuous everywhere except on a set of measure 0. So, the derivative has to be discontinuous on a set of positive measure. Now, to construct one of those!

Here's another favorite: For positive integer n and the set R of real numbers, suppose C is a closed subset of R^n with the usual topology. Then there exists a function f: R^n --> R that is 0 on C, strictly positive otherwise, and infinitely differentiable. So, for C, take, say, a sample path of Brownian motion, the Mandelbrot set, a Cantor set of positive measure, etc. Can use that function to settle an old question in constraint qualifications for the Kuhn-Tucker conditions in optimization.

Or, any closed set can be the level set of an infinitely differentiable function.

Sure, really fun reading for such things is:

Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964.

Fun book indeed!

Your first example you won't typically run into until an introductory measure theory course.

No, it's doable at the level of Rudin's Principles: He shows that a function is Riemann integrable if and only if it is continuous everywhere except on a set of measure 0.

For the definition of measure 0, he gives that quickly, and don't really need a course in measure theory. Besides a course in measure theory likes just to f'get about Riemann integration, and thankfully so.

Agree, it's do-able with baby Rudin. But as I recall it's a typical example used (early) in measure theory, not so much intro analysis. Hence "typical".

Ymmv, of course.

Thank you for pointing out my mistake! Indeed, that was a slip of mine to write about continuous functions, assuming they had derivatives.