1. Tardigrade
  2. Question
  3. Mathematics
  4. 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 :

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

JEE MainJEE Main 2022Continuity and Differentiability

Solution:

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