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.