Formal System Where Every Continuous Function is Almost Everywhere Differentiable Before Weierstrass defined his function, it was believed that every continuous function, $\mathbb{R} \to \mathbb{R}$, is almost everywhere differentiable.  Or at least, something approximating that theorem in the nomenclature of the time.  Many years later, Brouwer showed that, in his proposed system of constructive math, that every definable function was continuous (or maybe no function could be proved to be not continuous?).
Are there foundations for math where every continuous function is almost everywhere differentiable?  More generally, are there foundations for math where other fun conditions on functions always hold?
Thanks.
 A: Possibly work by Osvald Demuth (1936-1988) is an example (e.g. see this paper). See this google search and
Antonín Kučera, André Nies, and Christopher P. Porter, Demuth's path to randomness, Bulletin of Symbolic Logic 21 #3 (September 2015), pp. 270-305. 
17 July 2014 arXiv:1404.4449 version
Incidentally, a related paper -- Пример конструктивной недифференцируемой монотонной функции [An example of a constructive nondifferentiable monotone function] (MR 42 #7839 and Zbl 182.01802) -- is not mentioned in the above survey paper, but it's mentioned in Kushner's Lectures on Constructive Mathematical Analysis (1984).
Regarding what people believed before the mid 1870s, see my answer to Is Kline right that Cauchy believed that continuous functions must be differentiable?. In particular, I think it's important to note that nearly everyone at that time probably did not make a distinction between (or even consider that there might be a distinction between) "nondifferentiable at each point" and "nondifferentiable at a dense set of points" (and the dense set would not have been distinguished between being countable and being uncountable).
