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

To determine which of the given statements about the function \( f(x) \) is true, we need to analyze its continuity for each interval and check how the parameters \( a \), \( b \), and \( c \) affect it. 

The function \( f(x) \) is defined as:

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

**Analyzing Continuity:**

1. Interval \( (-\infty, 0) \): Here \( f(x) = \left[e^x\right] \). Since \( e^x \) is continuous and an exponential function, \( \left[e^x\right] \) is stepwise and thus discontinuous at integer values.
2. Interval \( [0, 1) \): Here, \( f(x) = \left[a e^x + \left[x-1\right]\right] \). The continuity depends on \( a \) and the behavior at both endpoints (\( x = 0 \) and \( x = 1 \)).
3. Interval \( [1, 2) \): Here, \( f(x) = \left[b + \left[\sin(\pi x)\right]\right] \). The term \( \left[\sin(\pi x)\right] \) may cause discontinuity at points where \( \sin(\pi x) \) changes sign.
4. Interval \( [2, \infty) \): Here, \( f(x) = \left[\left[e^{-x}\right] - c\right] \). Due to the exponential decay of \( e^{-x} \), discontinuities can occur at integer decreases of the exponent value.

**Checking the Options:**

• Option 1: "There exists \( a, b, c \in \mathbb{R} \) such that \( f \) is continuous on \( \mathbb{R} \)." This is unlikely due to the inherent nature of the greatest integer function causing stepwise jumps.
• Option 2: "If \( f \) is discontinuous at exactly one point, then \( a + b + c = 1 \)." After testing various combinations of \( a, b, \) and \( c\), finding a singular point of discontinuity under the condition that \( a + b + c = 1 \) is constrained.
• Option 3: "If \( f \) is discontinuous at exactly one point, then \( a + b + c \neq 1 \)." This arises because achieving continuity at certain junctures like \( x = 0, 1, \) and \( 2 \) while maintaining only one discontinuity depends on fine-tuning \( a, b, \) and \( c \).
• Option 4: "\( f \) is discontinuous at least two points, for any values of \( a, b, \) and \( c \)." This is generally true due to the behavior of the greatest integer functions involved, making continuity hard to maintain across multiple intervals.

**Conclusion:** The correct answer is that if \( f \) is discontinuous at exactly one point, then \( a + b + c \neq 1 \). This reflects the challenge in balancing the step functions over the defined intervals.