Why study $\mathcal{C}[0,1]$ I know this is probably a naive question here on this site, but I don't do a lot of higher math.  The group $\mathcal{C}[0,1]$ seems to be an important set within the framework of many mathematical texts.  I have not studied this outright, save for my Calc classes and very basic analysis.  I know there are segues into linear algebra with it as well, but why study this set of continuous functions?  What can groups of functions between the unit interval tell us about anything?  What behavior can be identified by studying them?  Is the most important framework for theory of measures and probability, considering the importance of the unit interval in the context of probability?  
 A: Here are some reasons to study this (in no particular order):
1) If you are learning stuff about normed linear spaces, the space of continuous functions with some norm (there are a lot of norms you can put on it, including but not limited to the sup norm, $||f||_{\infty}=\sup{\{|f(x)| | x\in{[0,1]}\}}$ and the $p$-norms $||f||_{p}=(\int{|f|^{p}})^{\frac{1}{p}}$ for $\infty>p\geq{1}$) is a fairly easy example to deal with (it is an easy example to cut your teeth on, so to speak).
2) Uniform convergence of continuous functions is an important topic, and this corresponds to convergence in the sup norm (For example the Stone- Weistrass theorem: http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem is an example of this and these results are very useful in higher level math).
3) It is not complete under the $p$-norms (i.e. Cauchy sequences don't necessarily converge.(for example the real line is complete but the rationals are not))
4) There is view point one can adapt when learning Lebesgue integration and some related parts of functional analysis (In calc you learn Riemann integration, which turns out to not be a "strong enough theory" of integration for a higher level of math). The so called $L^{p}$ spaces (this is where functional analysis starts) can be viewed as an attempt build a complete space (see 3), out of the existing incomplete space. (A similar idea is used by people to build the real numbers as a completion of the rationals).
etc.
