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 constantWhat would be an example of a non-well-defined function? Just any function which isnt increasing over the interval?c(cis between 0 and 1) from x=1 to x=2. What would the inverse function atcbe? Well a function assigns one output to any input. Which number from x=1 to x=2 should the inverse function evaluated atcbe? There should be 1 answer but there is a whole interval.