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Consider the function

$f(x)=\begin{cases} \int_0^x \{5+ \mid 1-y \mid \} dy & \quad if \: x>2 \\ 5x+2 & \quad if \: x \leq 2 \end{cases}$

Then

  1. $f$ is not continuous at x=2
  2. $f$ is continuous and differentiable everywhere
  3. $f$ is continuous everywhere but not differentiable at x=1
  4. $f$ is continuous everywhere but not differentiable at x=2
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Ans:- A

f(2)= 12

RHL=  $\int_{0}^{2} {5 + |1-y|} dy$
= 5y + ( y - y$^{2}$/2) sign(|1-y|)
= [10 + [2-2]*(-1)] - [ 5*0 + ( 0 - 0)*1]
=10
Since F(2)≄RHL
F(x) is not continuous at 2.

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