Question

Let f: ℝ → ℝ be defined as
\(f(x) = \left\{   \begin{array}{ll}     [e^x] &  x < 0 \\     [a e^x + [x-1]] & 0 \leq x < 1 \\     [b + [\sin(\pi x)]] &  1 \leq x < 2 \\     [[e^{-x}] - c] &  x \geq 2 \\   \end{array} \right.\)
Where a, b, c ∈ 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 ∈ ℝ such that ƒiscontinuous on ∈ ℝ .
• B. If ƒ is discontinuous at exactly one point, then a + b + c = 1
• C. If ƒ is discontinuous at exactly one point, then a + b + c ≠ 1
• D. ƒ is discontinuous at atleast two points, for any values of a, b and c

Answer 1

The Correct Option is C

The correct answer is (C) : If ƒ is discontinuous at exactly one point, then a + b + c ≠ 1
\(f(x) = \left\{   \begin{array}{ll}     0 &  x < 0 \\     a e^{x}-1 &  0 \leq x < 1 \\     b &  x = 1 \\     b - 1 & 1 < x < 2 \\     -c &  x \geq 2 \\   \end{array} \right.\)
To be continuous at x = 0
a – 1 = 0
to be continuous at x = 1
ae – 1 = b = b – 1 ⇒ not possible
to be continuous at x = 2
b – 1 = – c
⇒ b + c = 1
If a = 1 and b + c = 1 then f(x) is discontinuous at exactly one point.