Let f:ℝ→ℝ be defined as f(x)=ex, x<0aex+[x−1], 0≤x<1b+[sin⁡(πx)], 1≤x<2e−x−c, x≥2 where a,b,c∈ℝ and [t] denotes greatest integer less than or equal to t. Then which of the following statements is true ?

f(x) is discontinuous at x=1 For continuous at x=0; limx→0- ex=limx→0+aex+[x−1]⇒ a=1 For continuous at x=2; limx→2-b+[sin⁡(πx)], =limx→2+e−x−c⇒b=-1-c ⇒b+c=1 ⇒a+b+c=2

Let f:ℝ→ℝ be defined as f(x)=ex, x<0aex+[x−1], 0≤x<1b+[sin⁡(πx)], 1≤x<2e−x−c, x≥2 where a,b,c∈ℝ and [t] denotes greatest integer less than or equal to t. Then which of the following statements is true ?

f(x) is discontinuous at x=1 For continuous at x=0; limx→0- ex=limx→0+aex+[x−1]⇒ a=1 For continuous at x=2; limx→2-b+[sin⁡(πx)], =limx→2+e−x−c⇒b=-1-c ⇒b+c=1 ⇒a+b+c=2

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Let $f : ℝ \to ℝ$ be defined as $f (x) =$ $\{\begin{cases} [e^{x}], x < 0 \\ {ae}^{x} + [x - 1], 0 \leq x < 1 \\ b + [\sin (πx)], 1 \leq x < 2 \\ [e^{- x}] - c, x \geq 2 \end{cases}$ 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 ?

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

There exits $a, b, c \in ℝ$ such that f is continuous on R.

If f is discontinuous at exactly one point, then a + b + c = 1

f is discontinuous at atleast two points, for any value of a, b and c

answer is C.

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Detailed Solution

$f (x) is discontinuous at x = 1 For continuous at x = 0; \underset{}{\lim_{x \to 0^{-}} [e^{x}] = \lim_{x \to 0^{+}} {ae}^{x} + [x - 1] \Rightarrow} a = 1 For continuous at x = 2; \lim_{x \to 2^{-}} (b + [\sin (πx)],) = \lim_{x \to 2^{+}} ([e^{- x}] - c) \Rightarrow b = - 1 - c \Rightarrow b + c = 1 \Rightarrow a + b + c = 2$