# Almost everywhere differentiable definition

I dont understand what is the almost everwhere differentiable, can you give an example and definition please?

## 4 Answers

Example:
Cantor's staircase.
It has derivative zero except on the Cantor set, which is a set of measure zero.

All differentiable functions are also almost everywhere differentiable. But not vice versa. For example $e^x$ is everywhere differentiable hence it is almost everywhere differentiable.

Take another example $f(x)=|x|$. This function is not differentiable at $0$ but it is differentiable in other point on $\mathbb{R}$. Hence this function is almost everywhere differentiable but not everywhere differentiable.

Last, a function may have infinitely many number of points, where it is not differentiable but it can still be almost everywhere differentiable. Because all points where the function is not differentiable add up to a measure which is zero. Have a look at Lebesgue measure.

An almost everywhere differentiable function is a function that is differentiable except on a set of measure zero, as @Alex Halm said.

Almost everywhere differentiability doesn't work out as nicely as everywhere differentiability. For instance, two function can have the same derivative almost everywhere, but differ by more than a constant.

For instance the "Devil's staircase" (the same as @GEdgar's Cantor's Staircase) is a function that has a derivative almost everywhere on the interval $$[0,1]$$, and that derivative is zero almost everywhere. However, it is an increasing function, which would contradict our intuition about derivatives being zero.

A special class of functions are absolutely continuous functions. Those are functions $$f:[a,b]\to \mathbb{R}$$ for which $$f'$$ exists almost everywhere, $$f' \in L^1([a,b])$$ and $$f(t)-f(a) = \int_a^t f'(\tau)d\tau$$

It means differentiable everywhere except on a set of measure 0