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I am not sure if this is true, but intuitively it seems that if a function is strictly increasing and it is also continuous...it is differentiable.

It may be because there are no bumps like in the absolute value.

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Not necessarily. Counterexample: $ f(x)=\begin{cases} x & \text{if }x<0,\\ 2x & \text{if }x\ge 0.\end{cases} $ Is continuous, strictly increasing but not differentiable at $x=0$.

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    @AsafKaragila A monotone function is differentiable almost everywhere.2012-04-12
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Take a look at Billingsley' Probability and Measure, example 31.1

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    Answers should be somewhat self-contained. References and links are fine, but descriptions and explanations in the answer help in case references are not available or links go stale.2013-01-30