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

Option: 1

There exists \mathrm{a, b, c \in \mathbb{R}}  such that \mathrm{f} is continuous on  \mathrm{\mathbb{R}} .


Option: 2

If \mathrm{f} is discontinuous at exactly one point, then \mathrm{a+b+c=1}


Option: 3

If \mathrm{f} is discontinuous at exactly one point, then \mathrm{a+b+c \neq 1}


Option: 4

 \mathrm{f} is discontinuous at atleast two points, for any values of \mathrm{a, b} and \mathrm{c}


Answers (1)

best_answer

\mathrm{f\left ( O^{-} \right )= 0}
\mathrm{f\left ( O^{+} \right )= a-1= f\left ( 0 \right )}

\mathrm{f\left ( 1^{-} \right )= a/e-1 ,\: \: f\left ( 1^{+} \right )= b-1,\: \: f\left ( 1 \right )= b.}
\mathrm{f\left ( 2^{-} \right )= b-1 ,\: \: f\left ( 2^{+} \right )= -c,\: \: f\left ( 2 \right ).}

So f has a discontinuity at \mathrm{x= 1.}
If f is continuous at  \mathrm{x= 0\: \Rightarrow \: a-1= 0\: \Rightarrow\: a= 1. }
f is continuous at  \mathrm{x= 2\: \Rightarrow \: b-1= -c\: \Rightarrow\: b+c= 1 }

So \mathrm{a+b+c= 2\, \neq 1.}

Option (C)
 

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mansi

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