The correct options are
B f(x) is not continuous at x=2
C f(x) is differentiable for all x∈R
For x>2,
f(x)=x∫0(5+|1−t|) dt =1∫0(5+|1−t|) dt+x∫1(5+|1−t|) dt
=1∫0(6−t) dt+x∫1(4+t) dt
=[6t−t22]10+[4t+t22]x1
=6−12+4x+x22−4−12
=x22+4x+1
∴f(x)=⎧⎨⎩5x+1, x≤2x22+4x+1, x>2
At x=2, f(2)=11
limx→2−f(x)=11
limx→2+f(x)=2+8+1=11
So f(x) is continuous at x=2.
f′(x)={5, x≤2x+4, x>2
At x=2
limx→2−f′(x)=5
limx→2+f′(x)=2+4=6
limx→2−f′(x)≠limx→2+f′(x)
So f(x) is not differentiable at x=2
Therefore f(x) is differentiable for all x∈R−{2}