Are the $L^\infty$ functions differentiable?

Question

Let us consider a function $f:[a,b]\rightarrow\mathbb{R}^n$. If $f\in L^\infty([a,b],\mathbb{R}^n)$, then $f$ is differentiable for all $x\in[a,b]$?

Can somebody give me good references where I can found smooth and differentiable properties of $L^\infty$ and $W^{1,\infty}$ spaces?

Answer 1

Not at all. The "functions" in $L^p$ spaces are defined as equivalence classes of functions (defined by changes on zero measure set). Those "functions" only have to be integrable, or in the case of $L^\infty$, (essentially) bounded. This means that you cannot even evaluate a typical $L^\infty$-function pointwise, and for sure most functions are not differentiable anywhere.

You can read about $L^\infty$ and Sobolev spaces in all standard books about functional analysis, I recommend also a look into Evans "Partial Differential Equations" e.g. for Sobolev's lemma that tells you that if a function is in a Sobolev space with high enough order, then it is also differentiable in the classical sense.

Answer 2

The answer depends on your definition of differentiation.

If we are talking about classic differentiability, then no, nothing guarantees that. Classic example $$f(x) = \begin{cases}1,&x \in [0,0.5]\\0,&x\in(0.5,1]\end{cases}$$

If we are talking about weak derivatives, then again the same counterexample works, $f$ is not differentiable in the weak sense.

However, if we are talking about differentiability in the sense of distributions, then yes; all functions in $L^\infty (a,b)$ belong to $D'(a,b)$ and hence have a derivative in the sense of distributions.

Answer 3

As pointed out by daw in the comments, the function $f=[-1,1]\rightarrow\mathbb{R}$ defined by $$ f(x)= \begin{cases} 0&x\leqslant 0, \\ 1&x>0, \end{cases} $$ is an example of an $L^\infty$ function which is not weakly differentiable (because its distributional derivative is the Dirac delta distribution).

For differentiability properties of $W^{1,p}(U)$ functions, I recommend the book "Measure Theory and Fine Properties of Functions, Revised Edition" by Evans and Gariepy. In particular, Theorem 4.5 which characterizes $W^{1,\infty}_{\mathrm{loc}}(U)$ as the space of locally Lipschitz continuous functions in $U$ (this implies in particular that functions in $W^{1,\infty}(U)$ are continuous), Theorem 3.2 (Radamacher's theorem), which shows that locally Lipschitz continuous functions are differentiable almost everywhere, and Theorem 4.21 on differentiability of $W^{1,p}_{\mathrm{loc}}(U)$ functions on almost every line.

Evans, Lawrence Craig; Gariepy, Ronald F., Measure theory and fine properties of functions, Textbooks in Mathematics. Boca Raton, FL: CRC Press (ISBN 978-1-4822-4238-6/hbk). 309 p. (2015). ZBL1310.28001.