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.
[Math] continuous and strictly increasing implies differentiable
analysis
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Best Answer
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 $x=0$.