A well defined function would have a unique value assigned to each input (i.e. the CDF must be 1-to-1). Since CDF's are non-decreasing, in order to be 1-to-1, this says the CDF must be strictly increasing.
Nothing implies differentiable. In fact, suppose your CDF was:
This is certainly continuous but not differentiable everywhere (@ 5).
Yes. It would not be well-defined if the CDF was not increasing over some region. The CDF would not be 1-to-1. Suppose your CDF was constant c (c is between 0 and 1) from x=1 to x=2. What would the inverse function at c be? Well a function assigns one output to any input. Which number from x=1 to x=2 should the inverse function evaluated at c be? There should be 1 answer but there is a whole interval.What would be an example of a non-well-defined function? Just any function which isnt increasing over the interval?