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In my intermediate microeconomics class last year, I was rather surprised by the math involved in building consumer theory. In consumer theory you do things like define a binary relation $\succsim_a$ over a set of outcomes $X$ where $x\succsim_ay$ represents the relation "consumer a weakly prefers good $x$ to good $y$", and then reasoned about that relation by looking at its properties—for instance, we looked at the proof that if $\succsim_a$ is continuous then there is a continuous function $u_a:X\rightarrow \mathbb{R}$ exists, or that if $\succsim_a$ is convex then there is always a most preferred outcome in $X$ under a budget constraint.

This was my first exposure to this kind of mathematics, and I don't even know where it comes from. It seems obviously, to me at least, derived from some area of mathematics, but I don't know what that area is. If I wanted to take a course in maths similar to this, what might I look for in the course title?

P.S. I'm not sure how to tag this.

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The math used in microeconomics is a bit eclectic, but most of the things you need fall under the banner of real analysis, broadly defined. There are books that focus specifically on the material you are likely to need. You might need more convex analysis and multivalued functions than you learn in standard real analysis courses. Books I can recommend are Real Analysis with Economic Applications by Ok and An Introduction to Mathematical Analysis for Economic Theory and Econometrics by Corbae, Stinchcombe, and Zeman. If you have enough time and like to do lots of problems, Foundations of Mathematical Economics by Carter is great.

There are also great online ressources, such as various notes by Border, or, more structured, Foundations Of Economic Analysis by Border and Kvasov. A good starting point for doing consumer theory formally are the lecture notes of Ariel Rubinstein.

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Rubinstein's lecture notes are what we used. They were by far the most dense reading I had ever had to do at the time, but also the most fun. –  crf Aug 31 '12 at 23:17
Incidentally, now the honour of the most dense reading I've ever had to do goes to Rubinstein again: A Course In Game Theory by Osborne and Rubinstein. –  crf Aug 31 '12 at 23:21
@crf Your instructors have good taste. –  Michael Greinecker Aug 31 '12 at 23:23
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Many of the methods used in microeconomic theory are ultimately based on what universities call analysis.

This includes issues such as convexity, optimisation, and elasticity. Marginal costs and demand are little more than derivatives.

Once you move from theory to practice, other forms of mathematics are needed, such as the use of statistical and numerical methods to deal with actual and uncertain data.

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