Let `f(x)={(0 , x "is irrational"),(2/(2q^(3)-q^(2)+q+sin^(2)q+5) , if x=p/q ("rational")):}` (where HCF `(p,q)=1,p,q,gt0`) and `f(x)` is defined `AAxgt0` then which of the following is/are incorrect?

Options

A. `f(x)` is continuous at each irrational in `(0,oo)`
B. `f(x)` is continuous at each rational in `(0,oo)`
C. `f(x)` is discontinuous at each rational in `(0,oo)`
D. `f(x)` is discontinuous for all `x` in `(0,oo)`

Answer

B, D

Explanation

Let `x=sqrt(3)`
`f(sqrt(3))=0`
`because sqrt(3)=1.732050807`……….
As the decimal part increase then in the expression `p/q, q` becomes very large
So, `2/(2q^(3)-q^(2)+q+sin^(2)q+t)to0`
Hence `lim_(xtosqrt(3))f(x)=0`
Thus, `f(x)` is continuous at each irrational