At 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.