Are there any applications of techniques in real analysis to theoretical computer science? I have looked far and wide for such applications and have mostly turned up short. I can find plenty of applications of topology and similar structures on countable (or uncountable) sets, but rarely do I actually find uncountable sets as the object of study by computer scientists, and therefore leading up to the need for techniques from analysis.
Here are two related courses:


Analytical methods in combinatorics and CS by Ehud Friedgut at University of Toronto (2009),
Analytical Methods in Combinatorics and Computer-Science by Irit Dinur and Ehud Friedgut at Hebrew University (2006).


Also check Ryan O'Donnell's notes for his book:


Analysis of Boolean Functions by Ryan O'Donnell.


and the links on the top right corner.