Let f: R R be defined as f(x)= [ ll [e^ x ], , , , , , , , , , , … - Vidyadip

MCQ

Let $f: R \rightarrow R$ be defined as

$f(x)=\left[\begin{array}{ll}{\left[e^{x}\right],} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,x<0 \\ a e^{x}+[x-1], \,\,\,\,\,\,\,\,\,0 \leq x<1 \\ b+[\sin (\pi x)], \,\,\,\,\,\,\,\,\,\,\,\,1 \leq x<2 \\ {\left[e^{-x}\right]-c,} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,x \geq 2\end{array}\right.$

where a,b,c $\in R$ and $[t]$ denotes greatest integer less than or equal to $t.$ Then, which of the following statements is true $?$

A
There exists $a,b,c$ $\in R$ such that $f$ is continuous of $R$.
B
If $f$ is discontinuous at exactly one point, then $a+b+c=1$
✓
If $f$ is discontinuous at exactly one point, then $a+b+c \neq 1$
D
$f$ is discontinuous at atleast two points, for any values of $a , b$ and $c$.

✓

Answer

Correct option: C.

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

c
$f ( x )$ is discontinuous at $x =1$

For continuous at $x =0 ; a =1$

For continuous at $x =2 ; b + c =1$

$a+b+c=2$

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