provide a concrete example of a nowhere differentiable function and prove it is indeed nowhere differentiable?

Answer 1

f(x) = x; x an element of the natural numbers.

I guess you want an example of a function with domain R that is nowhere differentiable, LOL.

The Dirichlet function is defined as f(x) = 0 if x is irrational and 1 if x is rational. This function is discontinuous everywhere, it is fairly easy to prove since between any two arbitrary irrational numbers there exists a rational number and vice versa. Therefore the limit as x approaches 'a' for any 'a' an element of R cannot be determined.

since this function is nowhere continuous, it is nowhere differentiable.

Answer 2

The indicator function of the rational numbers - e.g. f(x) = {1 if x∈Q, 0 if x∉Q}. This function is nowhere continuous, and thus a fortiori nowhere differentiable.

Answer 3

an ' iterating snowflake' function is continuous and nowhere differentiable.

Answer 4

in the equation x = 6 you couldn't find the derivative of y with respect to x.