Let f , g: R arrow R be functions defined by f(x)= begincases[x] , x

Question

Let f , g: R arrow R be functions defined by f(x)= begincases[x] , x<0 |1-x| , x ≥ 0 endcases and g(x)= begincasesex-x , x<0 (x-1)2-1 , x ≥ 0 endcases where [ x ] denote the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly : (A) one point (B) two points (C) three points (D) four points

Tardigrade Admin · Accepted Answer

Check continuity at x =0 and also check continuity at those x where g(x)=0 g ( x )=0 at x =0,2 textfog(0+)=-1 textfog(0)=0 Hence, discontinuous at x =0 textfog(2+)=1 textfog(2-)=-1 Hence, discontinuous at x =2

Q. Let $f, g : R \to R$ be functions defined by
$f (x) = {[x] ∣1 - x ∣, x < 0, x \geq 0$ and
$g (x) = {e^{x} - x (x - 1)^{2} - 1, x < 0, x \geq 0$
where $[x]$ denote the greatest integer less than or equal to $x$ . Then, the function fog is discontinuous at exactly :

one point

two points

three points

four points

Check continuity at $x = 0$ and also check continuity at those $x$ where $g (x) = 0$ $g (x) = 0$ at $x = 0, 2$
$fog (0^{+}) = - 1$
$fog (0) = 0$
Hence, discontinuous at $x = 0$
$fog (2^{+}) = 1$
$fog (2^{-}) = - 1$
Hence, discontinuous at $x = 2$

Q. Let f,g:R→R be functions defined by f(x)={[x]∣1−x∣​,x<0,x≥0​ and g(x)={ex−x(x−1)2−1​,x<0,x≥0​ where [x] denote the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly :