# Differentiable and non-differentiable at infinite points

1. Oct 27, 2013

### JanEnClaesen

Let N be a natural number, let f(x) be equal to 1 for 0 <= x <= 1/N and let f( ] (n-1)/N, n/N ] ) = -f( ] n/N, (n+1)/N ] ), where n is a natural number and smaller or equal to N and where the function is defined for every 0 <= x <= 1.

Basically this a function composed of oscillating constant functions.
f is differentiable for every x from ]0,1] / {1/N, ... , N/N}, as N approaches infinity, f is differentiable at #(Reals / Naturals) points and not-differentiable at #Naturals points.

For finite N, the definite integral of f(x) for [0,1] is equal to 0, -1 or +1. For infinite N, the limit of this definite integral is indetermined.

In other words, nothing earth-shattering, but is what I've said correct and perhaps there's more to say about this function?

Last edited: Oct 27, 2013