Let `f : [a,b]-> R` be any function which is such that `f(x)` is rational for irrational x and that `f(x)` is iirrational for rational x, then in [a,b]

Options

A. f is discontinuous everywhere
B. f is discontinuous only at x = 0 and discontinuous everywhere
C. f is continuous for all irrational x and discontinuous for rational x
D. f is continuous for rational x and discontinuous for irrational x

Answer

A

Explanation

`f(x)={{:("rational",if,x cancelinQ" in "[a,b]),("irrational",if,x inQ" in "[a,b]):}`
Let `c in Q`
`therefore" "f(x)="irrational"`
`underset(xrarrc)(lim)f(x)=underset(hrarr0)(lim)f(c+h)="rational or irrational"`
`therefore` f(x) is discontinuous for x = c, hence for all x in [a, b]