Error estimates for finite difference formulae for numerical derivatives, polynomial interpolation, and numerical quadratures are usually given under the hypothesis that the relevant function has continuous derivatives up to some order. For example, the asymmetric finite difference approximation for the first derivative can be written as $$ \frac{f(x+h)-f(x)}{h} = f'(x) + \frac12 f''(\xi)h $$ with $\xi \in (x,x+h)$, and, provided the function is $C^2$, the previous formula allows us to get easily an upper bound for the error.
Similarly, in the case of the linear interpolation through two points $x_1$, and $x_2$, under the assumption that the interpolated values are values of a function $f(x) \in C^2[x_1,x_2]$, we can quickly obtain an absolute value of the error proportional to $(x_2-x_1)^2|f''(\xi)|$.
Is there anything we can say in general under weaker assumptions on $f$? Do we pass abruptly from nice error estimates for smooth functions to a no man's land with no information about the errors?