What are the real-world applications of real analysis?

Engineers and physicists need to solve differential equations. Most differential equations are impossible to solve symbolically. Instead, one obtains a numerical solution by either approximation methods or infinite series. Analysis was invented to understand the circumstances under which these methods produce correct answers. For example, when does the formal power series solution of a differential equation converge to the actual solution?

Each time someone does a derivative, integrates something, or solves a differential equation she is doing real analysis. Our modern world (starting with the most down-to-earth apliances and even furniture) would be totally impossible without it.