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

Answer: (C)

If ƒ is discontinuous at exactly one point, then a + b + _c _≠ 1

Answer 2

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.