Question

Let $f:R\to R$ be defined as $f\left(x\right)=[\begin{array}{cc}\left[e^x\right],& x<0\\ a{e}^x+[x-1],& 0\le x<1\\ b+\left[\sin \right(\pi x\left)\right],& 1\le x<2\\ \left[e^{-x}\right]-c,& x\ge 2\end{array}$ where $a,b,c\in R$ and $\left[t\right]$ denotes greatest integer less than or equal to $t$. Then, which of the following statements is true?

Answer 1

$f\left(x\right)=\{\begin{array}{cc}0& x<0\\ a{e}^x-1& 0\le x<1\\ b& x=1\\ b-1& 1<x<2\\ -c& x\ge 2\end{array}$
To be continuous at $x=0,$
$a-1=0$
To be continuous at $x=1,$
$ae-1=b=b-1\Rightarrow$ not possible
to be continuous at $x=2$
$b-1=-c\Rightarrow b+c=1$
If $a=1$ and $b+c=1,$ then $f\left(x\right)$ is discontinuous at exactly one point.