Find the value of x where function ,
`f(x)={{:(x,"if x is rational "),( 1-x, " if x is irrational "):}"is continuous."`

Options

A. `oo`
B. 1
C. 0
D. none of these

Answer

C

Explanation

Let a be any rational number. Then, there exists a sequence `lt a_(n) gt` of irrational points such that lim `a_(n)=a`
We have,
`therefore underset(n to oo)lim f(a_(n))=1-a`
`Rightarrow underset(n to oo)lim f(a_(n))=1-f(a)" "[therefore f(a)=a]`
Thus, lim `a_(n)`=a but lim `f(a_(n)) ne f(a)`.
So,f is discontinuous at x=a
Similarly, f(x) is discontinuous at all iraational poitns.
Hence, f(x) is nowhere continuous