MA005: Calculus I

The graph of the greatest integer function has a break or jump at each integer value, but how many breaks can a function have? The next function illustrates just how broken or "holey" the graph of a function can be.

Define $h(x) =\left\{ \begin{array}{ll} 2 & \mbox { if x is a rational number }\\1 & \mbox { if x is an irrational number} \end{array} \right.$

Then $h( 3 ) = 2$, $h( 5/3 ) = 2$ and $h( –2/5) = 2$ since 3, 5/3 and –2/5 are all rational numbers. $h( π ) = 1, h(\sqrt 7 ) = 1$, and $h( \sqrt 2 ) = 1$ since $π, \sqrt7$ and $\sqrt 2$ are all irrational numbers. These and some other points are plotted in Fig. 16 .

In order to analyze the behavior of $h(x)$ the following fact about rational and irrational numbers is useful.


Fact: "Every interval contains both rational and irrational numbers" or, equivalently, "If $a$ and $b$ are real numbers and $a < b$, then there is

The Fact tells us that between any two places where the $y = h(x) = 1$ (because $x$ is rational) there is a place where $y = h(x)$ is 2 because there is an irrational number between any two distinct rational numbers. Similarly, between any two places where $y = h(x) = 2$ (because $x$ is irrational) there is a place where $y = h(x) = 1$ because there is a rational number between any two distinct irrational numbers. The graph of $y = h(x)$ is impossible to actually draw since every two points on the graph are separated by a hole. This is also an example of a function which your computer or calculator can not graph because in general it can not determine whether an input value of $x$ is irrational.


Example 7: Sketch the graph of $g(x) = \left\{ \begin{array}{ll} 2 &\mbox { if x is a rational number }\\ x & \mbox { if x is an irrational number } \end{array} \right.$

Solution: A sketch of the graph of $y = g(x)$ is shown in Fig. 17 .

When $x$ is rational, the graph of $y = g(x)$ looks like the "holey" horizontal line $y = 2$. When $x$ is irrational, the graph of $y = g(x)$ looks like the "holey" line $y = x$.

Practice 9: Sketch the graph of $r(x) = \left\{ \begin{array}{ll} 2 & \mbox { if x is a rational number }\\ x & \mbox { if x is an irrational number } \end{array} \right.$