A continuity curiosity

It’s sunday, and I don’t feel like writing lots of source code and posting it here today, so I’ll make a shorter post this time…

In physics and engineering, functions describing physical quantities are often classified as piecewise continuous, everywhere continuous, differentiable, continuously differentiable or complex analytical, which is a list of requirements of increasing strictness. In classical mechanics, most functions are continuous and “well-behaved”, excluding some fractal phase-space curves in chaotic systems.

In pure mathematics, there often happens some kind of play with badly discontinuous or in some other way “detached from reality” functions. Let me show one example of this kind of a function.

The continuity of a real function f:\mathbb{R}\rightarrow\mathbb{R} at point x_0 is defined by


or in other words


which practically means that if we set any error bars f(x_o)-\epsilon \leq f(x) \leq f(x_0)+\epsilon for f(x), no matter how strict, there are always some error bars x_0 - \delta \leq x \leq x_0 + \delta for x that will keep the function value between those error bars.

So, here’s an example curious function that is special in the sense that it is continuous at only one point:


which is a real function that has value x when x is a rational number and has value -x when x is an irrational number (the notation \mathbb{Q}^C means the complement of \mathbb{Q}).

Left as an exercise for the reader, is to explain how this function is continuous only at the point x=0.

Sometimes later, I’ll write about fractal curves, which are continuous everywhere but are nowhere differentiable and have no finite curve length.

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