I was thinking of a pathological function that, according to my intuitive ideas, would be discontinuous, but it actually satisfies a certain kind of continuity.(adsbygoogle = window.adsbygoogle || []).push({});

First I claim that any element x∈[0,1) can be expressed in its decimal [or other base] expansion as

x=0.d_{1}d_{2}d_{3}...

Where each d_{i}is an element from the set {0,1,2,3,4,5,6,7,8,9} [with easy generalizations to any other base].

Let me define a function

f:[0,1)→[0,1)×[0,1)

x=0.d_{1}d_{2}d_{3}d_{4}d_{5}d_{6}...→(0.d_{1}d_{3}d_{5}..., 0.d_{2}d_{4}d_{6}...)

Now I'm just really curious about this function. Such a mapping must "jump around" to my mind. It has no derivative of course, but I think it's worse than the famous nondifferentiable ℝ→ℝ functions like the Wierstrauss function since it isn't even composed of continuous functions.

But if we take a typical Euclidean norm for points in [0,1)×[0,1),

|(x_{1},y_{1})-(x_{2},y_{2})| = [(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}]^{1/2}

then we can use the ε-δ definition and say:

For all x, x'∈[0,1)

|x-x'|=(x-x')<10^{-n}⇒ |f(x)-f(x')|<√2 * 10^{-n/2}.

So it is continuous.

Is this right? It is similar to the Weierstrauss function in that it is nondifferentiable but continuous, but it seems like its continuity must be of a different sort. Is there any classification for this kind of function, or are there any interesting rules of thumb for understanding these pathological functions? Is this particular function of any use, perhaps since it is idiosyncratically continuous?

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# Continuity of functions from ℝ→ℝ[sup]2[/sup]

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