Question

Do the lectures specify at any point the definition of a 'Differentiable function'? The concept was used in a example and a recitation session..

Answer 1

A "differentiable function" is one which is continuous over all the values of x you're interested in, and its derivative is also continuous. This means there are no sharp corners in the original function, only smooth transitions.

Answer 2

Differentiation is defined by a limit, and limits need not always exist.

If the limit \[\lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}\] exists at a particular x value then we say the function is differentiable at that point. It it exists for all x in an interval then it is differentiable on that interval.

Note that the limit is double sided, so implicit in this definition is the fact that no function defined on a closed interval can be differentiable on that closed interval, thus why the wording of many of the basic theorems insist on certain intervals being open...