Chapter 2.7, Problem 58E

Proving that $\underset{x\to a}{\lim }f(x)\ne L$ Use the following definition for the nonexistence of a limit. Assume f is defined for all values of x near a, except possibly at a. We write $\underset{x\to a}{\lim }f(x)\ne L$ if for some ε > 0, there is no value of δ > 0 satisfying the condition

$|f(x)-L|<\epsilon \text{whenever}0<|x-a|<\delta .$

58. Let

$f(x)=\{\begin{array}{ll}0\hfill & \text{if }x\text{ is rational}\hfill \\ 1\hfill & \text{if }x\text{ is irrational}\text{.}\hfill \end{array}$

Prove that $\underset{x\to a}{\lim }f(x)$ does not exist for any value of a. (Hint: Assume $\underset{x\to a}{\lim }f(x)=L$ for some values of a and L and let $\epsilon =\frac{1}{2}$.)