Question

Let $f\left( x \right)=\left\{ \begin{matrix}
   x,\text{ if }x\text{ is irrational} \\
   0,\text{ if }x\text{ is rational} \\
\end{matrix} \right.$, then f is
(a) Continuous everywhere
(b) Discontinuous everywhere
(c) continuous only at $x=0$
(d) continuous at all rational numbers

Answer 1

Hint: We start solving the problem by assuming a value of x as a rational number. We then find the value of the function at a rational number. We then find the left-hand and right-hand limits of the function at the assumed value for x. We then check whether the obtained left-hand and right-hand limits are equal to the value of function at that value to know about the continuity of the given function.

Complete step by step answer:
According to the problem, we are asked to find whether the given function $f\left( x \right)=\left\{ \begin{matrix}
   x,\text{ if }x\text{ is irrational} \\
   0,\text{ if }x\text{ is rational} \\
\end{matrix} \right.$ is continuous or not.
Let us assume a rational number R, then we get $f\left( R \right)=0$ ---(1).
Now, let us find the left and right-hand limits at $x=R$.
So, we have left-hand limit as $\displaystyle \lim_{x \to {{R}^{-}}}f\left( x \right)=\displaystyle \lim_{x \to {{R}^{-}}}x$.
$\Rightarrow \displaystyle \lim_{x \to {{R}^{-}}}f\left( x \right)=R$ ---(2).
So, we have right-hand limit as $\displaystyle \lim_{x \to {{R}^{+}}}f\left( x \right)=\displaystyle \lim_{x \to {{R}^{+}}}x$.
$\Rightarrow \displaystyle \lim_{x \to {{R}^{+}}}f\left( x \right)=R$ ---(3).
We know that if a function $f\left( x \right)$ is continuous at $x=a$, then $\displaystyle \lim_{x \to {{a}^{-}}}f\left( x \right)=\displaystyle \lim_{x \to {{a}^{+}}}f\left( x \right)=f\left( a \right)$.
From equations (1), (2) and (3), we can see that $\displaystyle \lim_{x \to {{R}^{-}}}f\left( x \right)=\displaystyle \lim_{x \to {{R}^{+}}}f\left( x \right)\ne f\left( R \right)$, which makes that the given function is discontinuous at every rational value of x which means that the function is discontinuous everywhere.

So, the correct answer is “Option b”.

Note: We should keep in mind that there will be infinite irrational numbers present between two rational numbers which makes our function not continuous. Whenever we get this type of problem, we first try to find the left and right-hand limits and then check whether it is equal to the value of function at that value of x. Similarly, we can expect problems to find the differentiability of the given function.