# Mathematics Weblog

## Two Interesting Functions

Sunday 23 May 2004 at 10:33 am | In Articles | 2 CommentsAt A level students only see reasonably nice functions, though occasionally they come across

where the maximum is smaller than the minimum (which, incidentally, is why one should use the terms *local maximum* and *local minimum* instead).

But what they won’t have come across are functions like these:

When trying to draw the graphs remember that

- between every pair of irrational numbers there is a rational number
- between every pair of rational numbers there is a irrational number

You can very roughly draw f using dots and there’s a nice picture of g at mathworld

Now think about integrating f and g – do they have areas under between them and the x-axis? The answer to ths question is important in the theory of integration.

These functions have other interesting properties and illustrate the relationship between rational and irrational numbers.

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Nice site. I love such functions.

Comment by Alex — Tuesday 25 May 2004 7:50 pm #

I just came across those functions f and g in Spivak’s “Calculus”. Trying to plot them both was actually quite fun and the result is interesting.

Comment by Tom — Sunday 24 October 2004 2:09 am #