Are there any applications of techniques in real analysis to theoretical computer science?
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.