The Dirichlet function defined as

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

A. Continuous for all real x

B. Continuous only at some values  of x

C. Discontinuous for all real x

D. Discontinuous only at some values of x

Hint: To solve this question we have to take an arbitrary rational number and check it’s continuity and also take an arbitrary irrational number and also check it’s continuity.

Complete step-by-step answer:
Let x’ be any arbitrary real number.
Case 1. X’ is rational
Then, f(x’) = 1
Because it is given in question if any real rational number will be input then output of function will be 1.
If any vicinity of a rational point there are irrational points, where f(x) = 0.
Hence, in any vicinity of x’ there are points x for which
|Δy|=|f(x′)−f(x)|=1 .
Therefore, x’ is a point of discontinuity.
Since x’ is an arbitrary point. the dirichlet function f(x) is discontinuous at each point.
Hence option (C) is the correct option.

Note: Whenever we get this type of question the key concept of solving is we should have knowledge of checking continuity and also understand the formula of checking continuity at that point. In the formula of continuity if it tends to 0 that means continuous otherwise discontinuous.