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?



Why are there so many smooth functions?
Given a paracompact smooth manifold, you have smooth partitions of unity ([nLab](http://ncatlab.org/nlab/show/partition+of+unity)), but on a real analytic manifold (e.g. a complex manifold viewed as a real manifold) one doesn't have analytic partitions of unity (much less holomorphic, if you are in the complex case). That is, given any open cover on a smooth manifold, one can find a partition of unity subordinate to that cover - this is a very topological property. Using partitions of unity you can paste together local functions as desired.


The existence of smooth partitions of unity comes down to the existence of a smooth (but not analytic!) bump function on $[-1,1]$. **Edit:** you can find details and formulas [on Wikipedia](https://en.wikipedia.org/wiki/Non-analytic_smooth_function#Smooth_transition_functions).


A related fact is that on a paracompact smooth manifold, the sheaf of real-valued functions is fine ([nLab](http://ncatlab.org/nlab/show/fine+sheaf),[wikipedia](http://en.wikipedia.org/wiki/Fine_sheaf#Fine_sheaves)).


