Let $f: \mathbb{R} \rightarrow \mathbb{R}$ 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 $a, b, c \in \mathbb{R}$ and $[t]$ denotes greatest integer less than or equal to $t$. Then, which of the following statements is true?
Select the correct option:
A
There exists $a, b, c \in \mathbb{R}$ such that $f$ is continuous of $\mathbb{R}$.
B
If f is discontinuous at exactly one point, then a + b + c = 1.
C
If $f$ is discontinuous at exactly one point, then $a+b+c \neq 1$.
D
f is discontinuous at least two points, for any values of a, b and c.
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
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
Hello 👋 Welcome to Competishun – India’s most trusted platform for JEE & NEET preparation. Need help with JEE / NEET courses, fees, batches, test series or free study material? Chat with us now 👇
