In the real domain, are there any theorems or definitions that state all functions are differentiable?

Question

I want to ask about basic theory of calculus, say differentiation.

We know that not every function can be integrable, but as far as I know all functions are differentiable in the real domain.

My question: Are there any theorems or definitions that state that all functions are differentiable in real domain?

If there is any, can you state it or prove it?

Thanks.

Answer 1

When I arrived at university, my professor of mathematical analysis (twenty years ago, in Italy, the graduate program in mathematics used to have no calculus course at all, but directly mathematical analysis; we used Rudin's book) told us that the generic high school student believes that every function is of class $C^\infty$. Of course now you know that there are so many singular (i.e. non-differentiable) functions around you, but nevertheless I want to tell you the the most famous mathematicians who lived two centuries ago did believe that every function had a derivative.

The reason is that they - and probably you- thought that functions were elementary formulae like polynomials, trigonometric functions, logarithms and so on. These elementary functions are differentiable (up to some really unnatural cases) at all points of their domains of definition. Unluckily calculus courses teach us to deal with functions like everything was allowed: differentiating, integrating, finding inverses, etc.

So no, you can't prove in any way that any function is differentiable because that would be a wrong theorem.

Answer 2

Not every function is differentiable. Say, $f(x) = 0$ if $x<=0$ and $f(x) = 1$ if $x>0$. $f$ isn't differentiable at 0. Or, take a look at the Weierstrass function, which is continuous everywhere and differentiable nowhere.

Answer 3

Funny point of view:

We know that not every function can be integrable, but as far as I know all functions is differentiable in the real domain.

You know that not every function is integrable but you think that every function is differentiable? Have you every tried to differentiate a non-integrable function? You won't find any, because differentiability implies continuity implies integrability (at least on a compact domain).

So to find a function which is not differentiable, just take a function which is not integrable.