Are absolutely continuous functions analytic?

Question

I am asking for a proof that any absolutely continuous function with absolutely continuous derivatives is analytic, once I am studying a function with the first property and 'd like to obtain the second one.

I know that exist smooth functions that are not analytical, but and if the derivatives are absolutely continuous?

If I am wrong, a counterexample is really welcome.

Thank you.

Answer 1

A famous counterexamples can be obtained by studying the function

$$f(x)=\begin{cases} e^{-\frac{1}{x}} & \text{ when }x>0, \\ 0 &\text{ when }x\leq 0.\end{cases}$$

This function happens to be $\mathcal{C}^\infty(\mathbb{R})$ (this requires some work but nothing beyond calculus)

Then, the function $\phi (x)=f(1-x)f(x+1)$ is also in $\mathcal{C}^\infty(\mathbb{R})$, is nonnegative and vanishes outside $(-1,1)$. Hence, $\phi$ along with all its derivatives, is absolutely continuous. This function however is not analytic!

This kind of functions (some variations of them rather) are called smooth mollifiers. This is also a related Wikipedia article.

Answer 2

You are asking whether any absolutely continuous function, all of whose derivatives exist and are absolutely continuous, is analytic. This is false. Counterexample: any smooth, nonanalytic function.

Indeed, any smooth function on an interval $(a,b)$ is absolutely continuous (on any closed subinterval $[c,d]$ of $(a,b)$), and all its derivatives exist and are also absolutely continuous.

This is because if $f: [c,d] \rightarrow \mathbb R$ is continuous and differentiable on $(c,d)$, with continuous derivative, then $f$ is already absolutely continuous.

Answer 3

Posting the comment above as answer, as requested.

The function $$g(x) = \frac{1}{2} x|x|$$ is differentiable everywhere, with $$g'(x) = |x|$$ absolutely continuous. However, $g$ is not analytic: if $g$ is analytic, then $g'$ is differentiable. So this $g$ is a counterexample.

An absolutely continuous function (which might not be differentiable) in general is far from analytic. Even when a function is infinitely differentiable, it might not be analytic as the other answer shows.