expansion of a function

Consider a ``neat'' function such as what an engineer is most likely to use in his typical theory/work. Such a function would typically be: (i) defined over a single finite interval, (ii) continuous throughout, and (iii) smooth enough. In other words, the sort of a function they used when they introduced the very idea of a graph of a function to you, back in high-school. ... Feel free to add any other qualifications you like, but note them explicitly, e.g., (iv) bounded throughout, and (v) periodic.