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Differentiable and non-differentiable at infinite points

  1. Oct 27, 2013 #1
    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
  2. jcsd
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