There are many examples of functions that are differentiable but not absolutely continuous. But these examples are unbounded oscillating functions (see for example some of the answers to this question Differentiable but not Absolutely continuous). Of course we also know that absolute continuity implies differentiability almost anywhere, but not necessarily differentiability. On the other hand, every cumulative probability distribution function is bounded in the interval $[0,1]$, it is increasing and continuous on the right. The question is whether there is any cumulative probability distribution function that is differentiable at every point $x$ but is not absolutely continuous. If it exists, this would also be an example of a continuous distribution function without a density function.
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