Calculus How To

Dirichlet Function

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The Dirichlet function is defined as:


It’s a good example of a function with unusual limit behavior. At first glance, you might think that the function looks fairly simple, because it appears to neatly oscillate between zero and one. However, despite this apparent simplicity, it can’t easily be perceived. That’s because there are an infinite number of irrational numbers between every rational number. These rapid oscillations mean that the function has no limit at any real number, which also means that it isn’t a continuous function anywhere.



Modified versions of the Dirichlet function include the Popcorn Function.

The Impossible Dirichlet Function Graph

The problem of drawing a graph of the Dirichlet function stems from the fact that between every two rational numbers there is some irrational number and between every irrational number there is some rational number. Therefore, it’s impossible to draw the function’s graph because of the infinite number of jumps between 0 and 1 (Prestini, 2013). You can make a reasonable representation, which looks like lines:
dirichlet function

The graph, drawn like this, appears to fail the vertical line test, which would make you conclude it isn’t a function. That obviously is not correct, and the issue is that if you zoom in on the graph, it’s actually a series of very, very dense dots.

Graphs of Functions and the Vertical Line Test

vertical line test

If a graph passes the vertical line test (top) it’s a function. If not (bottom), then it isn’t a function.

The lack of a graph for the Dirichlet function is actually a very important concept in calculus. When you first start studying calculus, you’re introduced to the vertical line test to determine whether you have a function or not. However, the ability to draw a graph (or not) actually has nothing to do with the definition of a function: if a graph exists, then you can apply a vertical line test (which makes it fairly easy to spot a function). But if you can’t draw a graph (as with the Dirichlet function) that doesn’t mean it isn’t a function. All it means is that you can’t apply the vertical line test.

References

Larson, R. & Edwards, B. (2008). Calculus of a Single Variable. Cengage Learning.
Prestini, E. (2013). The Evolution of Applied Harmonic Analysis: Models of the Real World. Birkhäuser Boston.

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Stephanie Glen. "Dirichlet Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/dirichlet-function/
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