TITLE: Motivation for the rigour of real analysis
QUESTION [117 upvotes]: I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really enjoyed it so far.
One thing I feel I am lacking in is motivation. That is, the difference in rigour between the usual introduction to calculus class and real analysis seems to be quite strong. While I appreciate rigour for aesthetic reasons, I have trouble understanding why the transition from the  18th century Euler style calculus to the rigorous "delta-epsilon" formulation of calculus was necessary. 
Is there a book that provides some historical motivation for the rigorous developement of calculus? Perhaps something that gives several counterexamples that occur when one is only equipped with a non-rigorous (i.e. first year undergraduate) formulation of calculus. For example, were there results that were thought to be true but turned out to be false when the foundations of calculus were strengthened? I suppose if anyone knows good counterexamples themselves they could list them here as well.

REPLY [2 votes]: @TheGreatDuck This is a fascinating thread. Let me comment on your aeronautical contribution from the viewpoint of an aeronautical engineer.
There are times when rigor is important and times when it isnt. For the first situation, consider the design of software to undertake air traffic control. Much attention is being given at the moment to "verifiable" algorithms, where it can be "rigorously" established that no possible situation has been overlooked. I am not sure that the standard of rigor would convince a modern analyst, but there is a recognition that intuition can be misleading and that formal analysis has considerable value.
An example where the search for rigor would be misplaced would be the calculation of the airflow by solving the Navier-Stokes equations. Insistence on rigor would require waiting around until the Navier-Stokes equations are shown to be well-posed, which is probably not going to happen soon. Until that day comes, designers will rely on wind-tunnel experiments, flight tests, and decades of accumulated experience. For now, this is MUCH safer than attempting to prove theorems. In fact, if I knew that the designers were trying to rely on theorems I would think very seriously before buying an airline ticket.
The value of rigor depends entirely on what you are trying to do, and how quickly you need to do it. This is true within mathematics as much as in its applications. Without Euler's gleeful nonchalance the pace of mathematical advance would have been greatly delayed.