Along the same lines as Qiaochu's and Zach's responses, the commonly taught heuristics pertaining to functions, differentiability and integration are a pet hate of mine.
I certainly left school thinking of functions as formulas involving combinations of elementary functions and having a very poor understanding of the relevance and correct relationship between integration and differentiation, the worst manifestation of which, now that I'm a bit older, seems to have been that
Differentiation is a nice, computable operation and tells you about functions; integration is hard and tells you about areas under curves.
Areas under curves never seemed interesting. As an analyst, my personal feelings towards them are now almost entirely reversed and I think of integration as my friend and differentiation as the enemy.
Differentiation uses up regularity; integration smooths.