There's a quote by Poincare on the "new functions", such as continuous functions without derivatives, that were appearing during the second half of the 19th century. The fullest version I've found online is in this comment by John Baez on the n-Category Café. A shorter version is on the MacTutor's function concept page, which attributes the year of the quote to 1899.
Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner.
In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that.
If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum.
Does anyone know the source?