TITLE: Why are there so many smooth functions?
QUESTION [46 upvotes]: I do understand that my question might seem a little bit ignorant, but I thought about it a lot and still can't wrap my head around it. 
Analycity imposes very strong conditions on a map, from elementary ones like "locally zero implies globally zero", to a little bit more deep like the Hurwitz formula (in the complex case). Neither of above are true if we just assume smoothness. 
On one hand, it is quite easy to prove that smooth functions are dense in any "reasonable" function space (I guess it depends on what one considers reasonable, though…) - just convolve with smooth approximations of identity. Also, (although I do take it on faith), any map of two manifolds is homotopic to a smooth one and two homotopic smooth maps are actually smooth-homotopic. 
Because of above facts, it seems to me that smooth functions are "abundant" and are actually very close to topology, ie. mere continuity. 
On the other hand, when I recall basic calculus course, it always seemed like being differentiable even once is a "miracle", and being differentiable infinitely many times is a very, very strong condition, even more so in several variables. 
Why are objects so constrained, ie. smooth functions, so useful and also, so malleable?

REPLY [6 votes]: On the other hand, in the Banach space $C[0,1]$ with the supremum norm, the set of functions which do not have (finite) derivative at any of the points of $[0,1]$ is residual. This was first proved by Stefan Banach and Stefan Mazurkiewicz in 1930s. A proof following J. C. Oxtoby can be found here (if you read Spanish): 
http://www.ciens.ula.ve/matematica/publicaciones/libros/por_profesor/wilman_brito/Teorema_de_Baire_Aplicaciones.pdf
For an even more refined result (a combination of several ones), look up Banach-Mazurkiewicz-Jarnik theorem (Theorem 7.2.1) in the following book:
Marek Jarnicki, Peter Pflug: Continuous nowhere differentiable functions. Monsters of analysis. Springer Monographs in Mathematics, 2015.