Which of the following is True?

Options

A. if `f(x)` is continuous at `x=c` and `g(x)` is discontinuous at `x=c` then `(f.g)(x)` must be discontinuous
B. If `f(x)` is continuous at `x=c` and `g(x)` is discontinuous at `x=c` then `(f.g)(x)` may be continuous.
C. If `f(x)` and `g(x)` are discontinuous at `x=c`, then the product function must be discontinuous.
D. If `f(x)` and `g(x)` are discontinuous at `x=c`, then the product function may be continuous.

Answer

B, D

Explanation

(A) & (B) Let `f(x)=x,xepsilonR` & `g(x)={("sin"(pi)/x,x!=0),(0,x=0):}`
`f(x)` is continuous at `x=0` but `g(x)` is not.
Now, `(f.g)(x)={(x"sin"(pi)/x,x!=0),(0, "at" x=0):}`
is continuous at `x=0`
(C) & (D)
Let `f(x)={(0, xepsilonQ),(1,x!inQ):} , g(x) {(1,xepsilonQ),(0,x!inQ):}`
`f(x)` & `g(x)` both are discontinuous everywhere.
but `f(x).g(x)=0,xepsilonR`