It is a common question, "what is the use of real analysis", and the answer is usually "it adds rigour to our mathematical tools and machinery to make sure that they work without just saying they do". This is not exactly "conventionally useful" (by my definition that may not be the best) because the fact that rational numbers aren't dense or whatever doesn't really make cars go faster or light bulbs more efficient (or does it?). Is real analysis such a highly abstracted topic that it doesn't influence real life within say 3 steps?
For example: analysis -> topology -> quantum physics-y stuff -> quantum computing -> fast computers, is 4 steps.
Has the rigour of real analysis shown an inconsistency overlooked by intuition and handwaving that actually had an easy to see impact?
For example, lets say that there is a mathematical model of the strength of a concrete wall under certain conditions. This wall is intended to be resistant to earthquakes. Lets say that the smart engineers who built the wall were under the impression that all continuous functions are differentiable - an intuitive thought.Does the fact that there exists continuous functions that are nowhere differentiable show that there could be an earthquake that breaks this wall that wasn't accounted for?
If computer chip designers were unaware that a signal could be broken down into component sine waves would they produce a product with security holes?
These example are exaggerated to give an idea of the type of relation i'm looking for.
The Borsuk-Ulam theorem showing that there exists a place on the earth which is at the same temperature as diametrically across the world is pretty cool.